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US History/Road to Revolution

The French and Indian War
(The following text is from Wikipedia)

The French and Indian War (1754–1763) was the North American chapter of the Seven Years' War. The name refers to the two main enemies of the British, the royal French forces and the various American Indian forces allied with them. This conflict, the fourth such colonial war between the kingdoms of France and Great Britain, resulted in the British conquest of all of New France east of the Mississippi River, as well as Spanish Florida. France ceded control of French Louisiana west of the Mississippi to its Spanish ally, to compensate it for its loss of Florida. By the end of this war France kept only the tiny islands of Saint Pierre and Miquelon north of the Caribbean. These colonies today still allow France access to the Grand Banks.

In Great Britain and France, the North American theatre of the Seven Years' War war usually has no special name, and so the entire worldwide conflict is known as the Seven Years' War (or the Guerre de sept ans). The "Seven Years" refers to events in Europe, from the official declaration of war in 1756 to the signing of the peace treaty in 1763. These dates do not correspond with the actual fighting in North America, where the fighting between the two colonial powers was largely concluded in six years, from the Jumonville Glen skirmish in 1754 to the capture of Montreal in 1760.

Elsewhere the conflict is known by several names. In British North America, wars were often named after the sitting British monarch, such as King William's War or Queen Anne's War. Because there had already been a King George's War in the 1740s, British colonists named the second war in King George's reign after their opponents, and thus it became known as the French and Indian War. This traditional name remains standard in the United States, although it obscures the fact that American Indians fought on both sides of the conflict. American historians generally use the traditional name or the European title (the Seven Years' War), and have also invented other, less frequently used names for the war, including the Fourth Intercolonial War and the Great War for the Empire. Canadian francophones and English speakers both refer to it as the Seven Years' War (Guerre de Sept Ans) or the War of the Conquest (Guerre de la Conquête), as the war in which New France was conquered by the British and became part of the British Empire. This war was also known as the Forgotten War.

Reasons for war
The French and Indian War began less than a decade after France and Great Britain had fought on opposing sides in the European War of the Austrian Succession (1740–1748). One cause for the conflict was territorial expansion. Newfoundland's Grand Banks were fertile fishing grounds and coveted by both sides. Both sides also wanted to expand their territories for trapping furs to trade, and for other pursuits that aided their economic interests. Both the British and the French used trading posts and forts to claim the Ohio Country, the vast territory between the Appalachian Mountains and the Mississippi River, from the Great Lakes to the Gulf of Mexico. English claims resulted from royal grants with no definite western boundaries. La Salle had claimed the Mississippi River for France: its drainage area includes the Ohio River Valley. Both Great Britain and France took advantage of Native American factions to secure these claims, to protect their territories, and to keep the other from growing too strong.

A second cause was political & religious ideology. The English Protestant colonists feared papal influence in North America. New France was administered by French governors and Roman Catholic hierarchy. French missionaries included Armand de La Richardie. English history was told as a story of freedom from Catholic (i.e., foreign) influence. French control over North America represented a threat to Great Britain. In their turn, the French feared English anti-Catholicism, in a time when Catholics were still being persecuted under English law.

Céloron's expedition
In June 1747 the Governor-General of New France, the Marquis de la Jonquière, ordered Pierre-Joseph Céloron to lead an expedition to the Ohio Country to remove British influence from the area. Céloron was also to confirm allegiance of the Native Americans in the Ohio territory to the French crown.

Céloron's expedition consisted of 213 soldiers of the Troupes de la marine (French Marines) transported by twenty-three canoes. The expedition left Lachine on June 15, 1749, and two days later reached Fort Frontenac. It then continued along the shoreline of present-day Lake Erie. At Chautauqua Portage (Barcelona, New York), it moved inland to the Allegheny River.

The troop headed south to the Ohio River at present-day Pittsburgh, where Céloron buried lead plates engraved with the French claim to the Ohio Country. Whenever British merchants or fur-traders were encountered by the French, they were informed of the illegality of being on French territory and told to leave the Ohio Country. When the expedition arrived at Logstown, the Native Americans there informed Céloron that they owned the Ohio Country, and they would trade with the British, despite anything the French said.

Céloron continued the expedition. At its farthest point south, it reached the junction between the Ohio River and the Miami River, just south of the village of Pickawillany. Here lived the old Chief of the Miami tribe, whom Céloron called "Old Britain." When Céloron arrived at Pickawillany, he informed the elderly Chief of "dire consequences" of continuing to trade with the British. "Old Britain" ignored the warning. After this meeting, Céloron and his expedition began the trip home, reaching Montreal only on November 10, 1749. In his report, Céloron wrote: "All I can say is that the Natives of these localities are very badly disposed towards the French, and are entirely devoted to the English. I don't know in what way they could be brought back."

Langlade's expedition
On March 17, 1752, Governor-General de la Jonquière died. His temporary replacement was Charles le Moyne de Longueuil. It was not until July 1, 1752 that Ange Duquense de Menneville arrived in New France to take over the post.

In the spring of 1752, Longueuil dispatched an expedition to the Ohio River area. The expedition was led by Charles Michel de Langlade, an officer in the Troupes de la marine. Langlade was given 300 men, some French-Canadians, and others members of the Ottawa tribe. His objective was to punish the Miami of Pickawillany for continuing to trade with the British. At dawn on June 21, 1752, the war party attacked the British trading center at Pickawillany, killing fourteen people of the Miami nation, including "Old Britain." The expedition then returned home.

Marin's expedition
In the spring of 1754, Paul Marin de la Malgue was given command of a 2,000 man force of Troupes de la Marine and Aboriginals. His orders were to protect the Ohio from the British. Marin followed the route that Céloron had mapped out four years before. However, where Céloron had buried lead plates, Marin was constructing and garrisoning forts. The first fort that was constructed by Paul Marin was at Presque Isle (Erie, Pennsylvania) on Lake Erie's south shore. He then had a road built to the headwaters of Rivière aux Boeuf (now known as Waterford, Pennsylvania). Marin then constructed a second fort at Le Boeuf, designed to guard the headwaters of the Rivière aux Boeuf.

Tanaghrisson's proclamation
On September 3, 1753, Tanaghrisson (d. 1754), Chief of the Mingo, arrived at Fort Le Boeuf. One tradition states that Tanaghrisson hated the French because they had killed and eaten his father. Tanaghrisson told Marin, "I shall strike . . .", threatening the French. The show of force by the French had alarmed the Iroquois in the area. They sent Mohawk runners to William Johnson's manor in Upper New York. Johnson, known to the Iroquois as "Warraghiggey", meaning "He who does big business", had become a respected member of the Iroquois Confederacy in the area. In 1746, Johnson was made a colonel of the Iroquois, and later a colonel of the Western New York Militia.

At Albany, New York, there was a meeting between Governor Clinton of New York and Chief Hendrick, as well as other officials from a handful of American colonies. Chief Hendrick insisted that the British abide by their obligations and block French expansion. When an unsatisfactory response was offered by Clinton, Chief Hendrick proclaimed that the "Covenant Chain", a long-standing friendly relationship between the Iroquois Confederacy and the British Crown, was broken.

Dinwiddie's reaction
Governor Robert Dinwiddie of Virginia found himself in a predicament. Many merchants had invested heavily in fur trading in Ohio. If the French made good on their claim to the Ohio Country and drove out the British, then the Virginian merchants would lose a lot of money. Dinwiddie could not possibly allow the loss of the Ohio Country to France. In October 1753 he wrote a letter to the commander of the French forces in the Ohio Country, Jacques Legardeur de Saint-Pierre, demanding an immediate French withdrawal. To deliver it he delegated Major George Washington of the Virginia militia. Major Washington left for Fort Le Boeuf on the 31st of October, along with his interpreter Jacob Van Braam and several other men.

A few days later, Washington and his party arrived at Wills Creek (Cumberland, Maryland). Here Washington enlisted the help of Christopher Gist, a surveyor who was familiar with the area. They arrived at Logstown on November 24, 1753. At Logstown, Washington met with Tanaghrisson, who was angry over the French military encroachment upon his land. Washington convinced Tanaghrisson to accompany his small group to Fort Le Boeuf.

On December 12, 1753, Washington and his men reached Fort Le Boeuf. Commander Saint-Pierre invited Washington to dine with him that evening. Over dinner, Washington presented Saint-Pierre with the letter from Dinwiddie. Saint-Pierre was civil in his response, saying, "As to the Summons you send me to retire, I do not think myself obliged to obey it." The French explained to Washington that France's claim to the region was superior to that of the British, as René-Robert Cavelier, Sieur de a Salle (1643–1687) had explored the Ohio Country nearly a century earlier.

Washington's party left Fort Le Boeuf early on December 16, 1753. By January 16, 1754, they had arrived in Williamsburg, Virginia. In his report, Washington stated, "The French had swept south." They had constructed and garrisoned forts at Presque Isle, Le Boeuf and Venango.

War
The French and Indian War was the last of four major colonial wars between the British, the French, and their Native American allies. Unlike the previous three wars, the French and Indian War began on North American soil and then spread to Europe, where Britain officially declared war on France on May 15, 1756, marking the beginnings of the Seven Years' War in Europe. Native Americans fought for both sides, but primarily alongside the French (with one exception being the Iroquois Confederacy, which sided with the American colonies and Britain). The first major event of the war was in 1754. Lieutenant Colonel George Washington, then twenty-one years of age, was sent to negotiate boundaries with the French, who did not give up their forts. Washington led a group of Virginian (colonial) troops to confront the French at Fort Duquesne (present day Pittsburgh). Washington discovered the French troops at the Battle of Jumonville Glen (about six miles or ten kilometers North-West of soon-to-be-established Fort Necessity). In the ensuing skirmish, a French Officer, Joseph Coulon de Jumonville, was killed. Washington pulled back a few miles and established Fort Necessity. The French forced Washington and his men to retreat. Meanwhile, the Albany Congress was taking place as means to discuss further action.

Edward Braddock led a campaign against the French at Fort Duquesne in 1755. Washington was again among the British and colonial troops. Braddock employed European tactics -- bold, linear marches and firing formations -- and employed heavy cannon. This led to disaster at the Monongahela. The French and natives were heavily outmanned and outgunned. But they used superior tactics, taking cover behind trees and bushes to gun down and rout the British. Braddock was killed. Despite four close calls, Washington escaped unharmed and led the survivors in retreat. This stunning British defeat heralded a string of major French victories over the next few years, at Fort Oswego, Fort William Henry, Fort Duquesne, and Carillon, where veteran Montcalm famously defeated five times his number. The sole British successes in the early years of the war came in 1755, at the Battle of Lake George, which secured the Hudson Valley; and in the taking of Fort Beauséjour (which protected the Nova Scotia frontier) by Lieutenant Colonel Robert Monckton. A consequence of this last battle was the subsequent forced deportation of the Acadian population of Nova Scotia and the Beaubassin region of Acadia.

In 1756 William Pitt became Secretary of State of Great Britain. His leadership, and France's continued neglect of the North-American theater, eventually turned the tide in favor of the British. The French were driven from many frontier posts such as Fort Niagara, and the key Fortress Louisbourg fell to the British in 1758. In 1759, the Battle of the Plains of Abraham gave Quebec City to the British, who had to withstand a siege there after the Battle of Sainte-Foy a year later. In September of 1760, Pierre François de Rigaud, Marquis de Vaudreuil-Cavagnal, the King's Governor of New France, negotiated a surrender with British General Jeffrey Amherst. General Amherst granted Vaudreuil's request that any French residents who chose to remain in the colony would be given freedom to continue worshiping in their Roman Catholic tradition, continued ownership of their property, and the right to remain undisturbed in their homes. The British provided medical treatment for the sick and wounded French soldiers, and French regular troops were returned to France aboard British ships with an agreement that they were not to serve again in the present war.

Summary of the war in America In 1752 the French and their Native allies raided a trading outpost sited at modern day Cleveland rid the area of Pennsylvanians. In 1754 General George Washington attacked French soldiers and then became trapped in his poorly built, Fort Necessity in Great Meadows Pennsylvania and more than one-third of Washingtons's men shortly became casualties. Twenty-two year old Washington and his men surrendered and were allowed to leave back to Virginia. In July 1755, a few miles south of Fort Duquensne in Pennsylvania, the combined forces of French and Natives attacked British colonial troops that were preparing a to assault the fort. The aftermath that ensued would result in a British defeat and General Edward Braddock would be killed. Once London heard of this Britain declared war upon France and formally began the seven years war. After this the British feared that France would attempt to retake Nova Scotia and that the 12,000 French Nova Scotians would break their neutrality, so the British military forced around seven thousand French Nova Scotians from their homeland. This was history's first large-scale modern deportation, the French would be sent from Louisiana to the Caribbean and families would become torn apart. In July of 1758 The British had recaptured the fort at Loiusberg winning control of the St. Lawerence River. This would cut the major French supply route and open up more supply lines for the British. In the fall of 1758 the Shawnee and Delaware Natives accepted peace offerings from the British and the French abandoned Fort Duquesne. A decisive attack would happen in the fall of 1759 when General James Wolfe's forces defeated the French on the Plains of Abraham and thus taking Quebec. A year after this event the British would capture Montreal and the North American stage of the war would be over.

Outcome
Though most of the North American fighting ended on September 8, 1760, when the Marquis de Vaudreuil surrendered Montreal — and effectively all of Canada — to Britain (one notable late battle allowed the capture of Spanish Havana by British and colonial forces in 1762), the war officially ended with the signing of the Treaty of Paris on February 10, 1763. The treaty sealed France's loss of all its North American possessions east of the Mississippi except for Saint Pierre and Miquelon islands off Newfoundland. All of Canada was ceded to Britain. France regained the Caribbean islands of Guadeloupe and Martinique, which had been occupied by the British. The economic value of these islands to France was greater than that of Canada at the time, because of their rich sugar crops; and the islands were easier to defend. However, the British were happy to take New France: defense was not an issue, and they had many sources of sugar. Spain gained Louisiana, including New Orleans, in compensation for its loss of Florida to the British.

French Canada contained approximately 65,000 French-speaking Roman Catholic residents. Early in the war, in 1755, the British had expelled French settlers from Acadia. (Some of these eventually fled to Louisiana, creating the Cajun population.) Now at peace, and eager to secure control of its hard-won colony, Great Britain made concessions to its newly conquered subjects with the Quebec Act of 1774. The history of the Seven Years' War, particularly the siege of Québec and the death of British Brigadier General James Wolfe, generated a vast number of ballads, broadsides, images, maps and other printed materials, which testify to how this event continued to capture the imagination of the British public long after Wolfe's death in 1759. The European theatre of the war was settled by the Treaty of Hubertusburg on February 15, 1763. The war changed economic, political, and social relations between Britain and its colonies. It plunged Britain into debt, which the Crown chose to pay off with tax money from its colonies. These taxes contributed to the beginning the American Revolutionary War.

Battles and expeditions

 * United States

 * Battle of Jumonville Glen (May 28, 1754)
 * Battle of Fort Necessity, aka the Battle of Great Meadows (July 3, 1754)
 * Braddock Expedition (Battle of the Monongahela aka Battle of the Wilderness) (July 9, 1755)
 * Kittanning Expedition (climax September 8, 1756)
 * Battle of Fort Duquesne (September 14, 1758)
 * Battle of Fort Ligonier (October 12, 1758)
 * Forbes Expedition (climax November 25, 1758)

 * Province of New York
 * Battle of Lake George (1755)
 * Battle of Fort Oswego (August, 1756)
 * Battle on Snowshoes (January 21, 1757)
 * Battle of Fort Bull (March 27, 1756)
 * Battle of Sabbath Day Point (July 26, 1757)
 * Battle of Fort William Henry (August 9, 1757)
 * Attack on German Flatts (1757) (November 12, 1757)
 * Battle of Carillon (July 8, 1758)
 * Battle of Ticonderoga (1759)
 * Battle of La Belle-Famille (July 24, 1759)
 * Battle of Fort Niagara (1759)
 * Battle of the Thousand Islands, 16-25 August, 1760

 * West Virginia
 * Battle of Great Cacapon (April 18, 1756)

 * Canada
 * New Brunswick
 * Battle of Fort Beauséjour (June 16, 1755)

 * Nova Scotia
 * Battle of Louisburg (July 27, 1758)

 * Ontario
 * Battle of Fort Frontenac (August 25, 1758)
 * Battle of the Thousand Islands, 16-25 August, 1760

 * Quebec
 * Battle of Beauport (July 31, 1759)
 * Battle of the Plains of Abraham (September 13, 1759)
 * Battle of Sainte-Foy (April 28, 1760)
 * Battle of Restigouche, July 3-8, (1760)

 * Newfoundland
 * Battle of Signal Hill September 15, 1762

Proclamation of 1763
(The following text is taken from the Wikipedia article)

The Royal Proclamation of 1763 was issued October 7, 1763 by George III following Great Britain's acquisition of French territory in North America after the end of the Seven Years' War. The purpose of the proclamation was to make sure Britain could control its new territory for its The Proclamation in essence forbade Americans from settling or buying land west of the Appalachians. Colonists were angry because many already had land in that area. Additionally, the Proclamation gave the Crown a monopoly in land bought from Native Americans.

Native land
In the fall of 1763, a royal decree was issued that prohibited the North American colonists from establishing or maintaining settlements west of an imaginary line running down the crest of the Appalachian Mountains. The proclamation acknowledged that Native Americans owned the lands on which they were then residing and white settlers in the area were to be removed.

However, provision was made to allow specially licensed individuals and entities to operate fur trading ventures in the proscribed area. There were two motivations for this policy:

To avoid warfare with the Indians. Neither side evidenced any affection for the tribes, but Indian conflicts were very expensive, and the British hadn't yet deployed enough soldiers in the West to keep the peace. Some Indians welcomed this policy, believing that separation from the colonies would allow them to resume their traditions. Others realized that the proclamation would, at best, only provide breathing room before the next onslaught of settlers.

To concentrate colonial settlements on the seaboard where they could be active parts of the British mercantile system. British trade officials took it as a first priority to populate the recently secured areas of Canada and Florida (referring to the Treaty of Paris), where colonists could reasonably be expected to trade with the mother country. Settlers living west of the Appalachians would be highly self-sufficient and have little opportunity to trade with English merchants.

The reaction of colonial land speculators and frontiersmen was immediate and negative. They believed their fight in the recent war had been "rewarded" by the creation of a vast restricted native reserve in the lands they coveted. Most concluded that the proclamation was only a temporary measure: a number ignored it entirely and moved into the prohibited area. Almost from its inception, the proclamation was modified to suit the needs of influential people with interests in the American West, both high British officials and colonial leaders.

Beginning in 1764, portions of the Proclamation Line were adjusted westward to accommodate speculative interests. Later, in 1768, the first Treaty of Fort Stanwix formally recognized the surrender of transmontane lands claimed by the Iroquois.

The Proclamation of 1763 was a well-intentioned measure. Pontiac’s Rebellion had inflicted a terrible toll on the frontier settlements in North America and the British government acted prudently by attempting to avoid such conflict in the foreseeable future.

The colonists, however, were not appreciative and regarded the new policy as an infringement of their basic rights. The fact that western expansion was halted at roughly the same time that other restrictive measures were being implemented, made the colonists increasingly suspicious

Almost immediately, many British colonists and land speculators objected to the proclamation boundary, since there were already many settlements beyond the line (some of which had been temporarily evacuated during Pontiac's War), as well as many existing land claims yet to be settled. Indeed, the proclamation itself called for lands to be granted to British soldiers who had served in the Seven Years' War. Prominent American colonists joined with land speculators in Britain to lobby the government to move the line further west. As a result, the boundary line was adjusted in a series of treaties with Native Americans. The Treaty of Fort Stanwix and the Treaty of Hard Labor (both 1768) and the Treaty of Lochaber (1770) opened much of what is now West Virginia and Kentucky to British settlement.

Organization of new colonies
Besides regulating colonial expansion, the proclamation dealt with the management of newly ceded French colonies. It established government for four areas: Quebec, West Florida, East Florida, and Grenada. All of these were granted the ability to elect general assemblies under a royally appointed governor or a high council, which could then create laws and ordinances specific to the area in agreement with British and colonial laws. In the meantime, the new colonies enjoyed the same rights as native-born Englishmen, something that British colonists had been fighting over for years. An even bigger affront to the British colonies was the establishment of both civil and criminal courts complete with the right to appeal--but those charged with violating the Stamp or Sugar Act were to be tried in admiralty court, where the defendant was considered guilty until he or she could prove his or her innocence.

Legacy
The influence of the Royal Proclamation of 1763 on the coming of the American Revolutionary War (1775–1783) has been variously interpreted. Many historians argue that the proclamation ceased to be a major source of tension after 1768, since the aforementioned treaties opened up extensive lands for settlement. Others have argued that colonial resentment of the proclamation contributed to the growing divide between the colonies and the Mother Country.

In the United States, the Royal Proclamation of 1763 ended with the American Revolutionary War, because Great Britain ceded the land in question to the United States in the Treaty of Paris (1783). Afterwards, the U.S. government also faced difficulties in preventing frontier violence, and eventually adopted policies similar to those of the Royal Proclamation. The first in a series of Indian Intercourse Acts was passed in 1790, prohibiting unregulated trade and travel in Native American lands. Additionally, the U.S. Supreme Court case Johnson v. M'Intosh (1823) established that only the U.S. government, and not private individuals, could purchase land from Native Americans.

The Royal Proclamation continued to govern the cession of aboriginal land in British North America, especially Upper Canada and Rupert's Land. The proclamation forms the basis of land claims of aboriginal peoples in Canada – First Nations, Inuit, and Metis. The Royal Proclamation of 1763 is thus mentioned in Section Twenty-five of the Canadian Charter of Rights and Freedoms.

The Stamp Act and other Laws
In 1764, George Grenville became the British Chancellor of the Exchequer (minister of finance). He allowed customs officers to obtain general writs of assistance, which allowed officers to search random houses for smuggled goods. Grenville thought that if profits from smuggled goods could be directed towards Britain, the money could help pay off debts. Colonists were horrified that they could be searched without warrant at any given moment. Also in 1764, with persuasion from Grenville, Parliament began to impose several taxes on the colonists. The Sugar Act of 1764 reduced the taxes imposed by the Molasses Act, but at the same time strengthened the collection of the taxes. It also provided that British judges, and not juries, would try cases involving that Act.

The next year, Parliament passed the Quartering Act, which required the colonies to provide room and board for British soldiers stationed in North America; the soldiers would serve various purposes, chiefly to enforce the previously passed acts of Parliament.

Following the Quartering Act, Parliament passed one of the most infamous pieces of legislation: the Stamp Act. Previously, Parliament imposed only external taxes on imports. But the Stamp Act provided the first internal tax on the colonists, requiring that a tax stamp be applied to books, newspapers, pamphlets, legal documents, playing cards, and dice. The legislature of Massachusetts requested a conference on the Stamp Act; the Stamp Act Congress met in October that year, petitioning the King and Parliament to repeal the act before it went into effect at the end of the month, crying "taxation without representation."

The act faced vehement opposition throughout the colonies. Merchants threatened to boycott British products. Thousands of New Yorkers rioted near the location where the stamps were stored. In Boston, the Sons of Liberty, a violent group led by radical statesman Samuel Adams, destroyed the home of Lieutenant Governor Thomas Hutchinson. Parliament did indeed repeal the Stamp Act, but additionally passed the Declaratory Act, which stated that Great Britain retained the power to tax the colonists, even without substantive representation.

Believing that the colonists only objected to internal taxes, Chancellor of the Exchequer Charles Townshend proposed bills that would later become the Townshend Acts. The Acts, passed in 1767, taxed imports of tea, glass, paint, lead, and even paper. The colonial merchants again threatened to boycott the taxed products, reducing the profits of British merchants, who in turn petitioned Parliament to repeal the Townshend Acts. Parliament eventually agreed to repeal much of the Townshend legislation. But Parliament refused to remove the tax on tea, implying that the British retained the authority to tax the colonies despite a lack of representation.

In 1773, Parliament passed the Tea Act, which exempted the British East India Company from the Townshend taxes. Thus, the East India Company gained a great advantage over other companies when selling tea in the colonies. The colonists who resented the advantages given to British companies dumped British tea overboard in the Boston Tea Party in December of 1773.

The Boston Tea Party

In retaliation for the Boston Tea Party, Parliament passed the Coercive Acts, which were in the colonies known as the Intolerable Acts. Parliament reduced the power of the Massachusetts legislature and closed the port of Boston. Also, the Quartering Act was extended to require private individuals to lodge soldiers. Furthermore, Parliament allowed royal officials accused of crimes to be tried by a British, rather than a colonial, jury.

First Continental Congress
In order to debate a response to the Intolerable Acts, all American colonies except for Georgia sent delegates to the First Continental Congress at Philadelphia. The Congress met in September 1774 and issued a Declaration of Rights and Grievances. When the Congress adjourned, it stipulated another Congress would meet if King George III did not meet the demands of the Declaration. When the Second Congress did meet, the military hostilities of the Revolutionary War had already begun, and the issue of Independence, rather than a redress of grievances, dominated the debates.

Education
Literacy grew for both men and women during the 18th century. In New England and the Middle States, more middle-class girls were sent to school. However, as Science and the requirements for citizenship became more a part of education, girls were excluded from learning these topics.

Higher education continued to develop, with the 1746 opening of The College of New Jersey (later known as Princeton), and King's College (now Columbia) in 1754. All of these universities were meant exclusively for White men, though some of the colleges experimented by admitting Native Americans. In the public schools, vocational education expanded.

Though what was lost by failing to educate the underclasses was incalculable, we can gauge the lost possibilities through such individuals as Benjamin Banneker and Phillis Wheatley. Mr. Banneker, a self-educated free African-American, observed the stars, wrote his own almanac, and was one of the surveyors of what would later become the District of Columbia. Miss Wheatley, an African-born slave educated and freed by her mistress, wrote a remarkable volume of poems published in the year 1773. Most of these had been published in The Newport Mercury, edited by Benjamin Franklin's brother James.

Questions For Review
1. What were the reasons for the French and Indian War?

2. What was the strategy of General Braddock against the French at Fort Duquesne? What was the strategy of the defending French and Indian forces?

3. Examine the succession of acts imposed upon the American Colonists in the wake of the war, beginning with the Sugar Act. What was the intended purpose of each act? What was its actual effect?

title: US History/Road to Revolution


text:
US History/American Revolution

Background
The British forces might at first glance seem to have every advantage. At the outset of the War they had stocks of cannon and ammunition. The Colonists had single-shot rifles from local forges, guns which took time to load and could easily misfire or explode. When Washington took command of the army in 1775, he learned that there was only enough gunpowder to provide nine rounds of ammunition per man. The British had a large professional army drilled to a pitch like that of Ancient Rome, well-supplied with food, uniforms, and arms. But the American lack of training meant that they did not mass in the European style. Instead they relied on snipers, individuals hidden behind the trees who shot their bullet and then loaded again while their neighbors fired. They had learned this during the French and Indian War. Snipers helped strengthen the American odds.

Lexington and Concord
The British government commanded General Thomas Gage to enforce the Intolerable Acts and limit rights in Massachusetts. Gage decided to confiscate a stockpile of colonial arms located in Concord. On April 19, 1775, Gage's troops marched to Concord. On the way, at the town of Lexington, Americans who had been warned in advance by Paul Revere and others of the British movements made an attempt to stop the troops. No one knows which side fired the first shot, but it sparked battle on Lexington Green between the British and the Minutemen. Faced against an overwhelmingly superior number of British regular troops in an open field, the Minutemen were quickly routed. Nevertheless, alarms sounded through the countryside. The colonial militias poured in and were able to launch guerrilla attacks on the British while they marched on to Concord. The colonials amassed of troops at Concord. They engaged the British in force there, and they were able to repulse them. They then claimed the contents of the armory. The British retreated to Boston under a constant and withering fire from all sides. Only a reinforcing column with artillery support on the outskirts of Boston prevented the British withdrawal from becoming a total rout. The following day the British woke up to find Boston surrounded by 20,000 armed colonists, occupying the neck of land extending to the peninsula the city stood on.

The Battle of Bunker Hill
"Don't shoot until you see the whites of their eyes."

The action changed from a battle to a siege, where one army bottles up another in a town or a city. (Though in traditional terms, the British were not besieged, since the Royal Navy controlled the harbor and supplies came in by ship.) General Artemas Ward, the head of the Massachusetts militia, had the initial oversight of the siege. He set up headquarters at Cambridge, Massachusetts and positioned his forces at Charlestown Neck, Roxbury, and Dorchester Heights. The 6,000 to 8,000 rebels faced some 4,000 British regulars under General Thomas Gage. Boston and little else was controlled by British troops. General Gage countered the siege on June 17 by attacking the colonists on Breed's Hill and Bunker Hill. Although the British suffered tremendous casualties compared to the colonial losses, the British were eventually able to dislodge the American forces from their entrenched positions. The colonists were forced to retreat when many colonial soldiers ran out of ammunition. Soon after, the area surrounding Boston fell to the British. However, because of the losses they suffered, they were unable to break the siege of the city. Despite the early defeat for the colonists, the battle proved that they had the potential to counter British forces, which were at that time considered the best in the world.

The Last Chance For Peace
The Second Continental Congress adopted the Olive Branch Petition, a petition for peace, on July 5, 1775. The Congress affirmed its allegiance to the Crown. It was received in London at the same time as it heard of the Battle For Bunker Hill. The King refused to read the petition or to meet with its ambassadors. Parliament reacted by passing the Prohibitory Act, which banned trade with the colonies.

Battle For Boston
Despite the British access to the ships, the town and the army were on short rations. Salt pork was the order of the day, and prices escalated rapidly. While the American forces had some information about what was happening in the city, General Gage had no effective intelligence of rebel activities.

On May 25, 1775, 4,500 reinforcements and three new generals arrived in Boston Harbor. The fresh leaders were Major General William Howe and Brigadiers John Burgoyne and Henry Clinton. Gage began planing to break out of the city.

On July 3, 1775, George Washington arrived to take charge of the new Continental Army. Forces and supplies came in from as far away as Maryland. Trenches were built at Dorchester Neck, extending toward Boston. Washington reoccupied Bunker Hill and Breeds Hill without opposition. However, these activities had little effect on the British occupation.

In the winter of 1775– 1776, Henry Knox and his engineers under order from George Washington used sledges to retrieve sixty tons of heavy artillery that had been captured at Fort Ticonderoga. Knox, who had come up with the idea to use sledges, believed that he would have the artillery there in eighteen days. It took six weeks to bring them across the frozen Connecticut River, and they arrived back at Cambridge on January 24, 1776. Weeks later, in an amazing feat of deception and mobility, Washington moved artillery and several thousand men overnight to take Dorchester Heights overlooking Boston. General John Thomas fortified the area. The British fleet had become a liability, anchored in a shallow harbor with limited maneuverability, and under the American guns on Dorchester Heights.

When General Howe saw the cannons, he knew he could not hold the city. He asked that George Washington let them evacuate the city in peace. In return, they would not burn the city to the ground. Washington agreed: he had no choice. He had artillery guns, but did not have the gunpowder. The whole plan had been a masterful bluff. The siege ended when the British set sail for Halifax, Nova Scotia on March 17, 1776. The militia went home, and in April Washington took most of the Continental Army forces to fortify New York City.

Ethan Allen and Fort Ticonderoga
The British had considered Fort Ticonderoga a relatively unimportant outpost in a conflict which had up to then been mostly based in Massachusetts. However, a veteran of the French and Indian War, Ethan Allen, had his eye on the fort. Allen had built up a Vermont territorial militia, the Green Mountain Boys, until it was an effective fighting force. Vermont was claimed by the New York colony, but Allen wanted more independence. In April of 1775, Allen was surprised by a visit by Commander Benedict Arnold of the Connecticut Militia. Arnold announced that he had been commissioned to seize the cannons of Fort Ticonderoga. A heated discussion between the two concluded with the agreement that the two militias would combine to attack the fort. This was for the best, for both forces together were small, well short of brigade strength. On May tenth, the combined American forces captured the fort. They seized the arms, including the cannons, which were then hauled by oxen all the way to Boston.

Strengthening The Cause
Through the media available in that day, the Revolution promoted the idea of honorable men in revolt against tyranny. Newspapers in North and South published incendiary stories and inspiring engravings. The theater contributed dramatic outcries, including those of Mary Otis Warren. Songs were played and sung to rally flagging spirits.

Thomas Paine's 1776
In January of 1776, the Englishman Thomas Paine published the pamphlet Common Sense. This anti-monarchical publication encouraged American independence, using examples from the Bible and republican virtues to argue that kings were never good for any free state. In late 1776 he began printing his series of pamphlets, The American Crisis, calling soldiers to mass to the cause of the Revolution. The first of these pamphlets begins with the stirring words, "These are the times that try men's souls."

The Declaration of Independence

As military hostilities built up, the Second Continental Congress appointed George Washington as General of the Continental Army. Washington gave up his salary for the position all through the war. (As he was among the richest men in the colonies, he could afford this choice.) In June of 1776, the Second Continental Congress felt it needed a spur for separation from Great Britain. It appointed a Committee of Five to draft a declaration of independence: John Adams, Benjamin Franklin, Robert Livingston, Roger Sherman and Thomas Jefferson. Jefferson became the principal author of this document.

Although the British king was no longer principally responsible for his dominion's policy, the Declaration of Independence called him a tyrant. It justified the rights of the rebellion with words the European Enlightenment would have hailed: "We hold these truths to be self-evident, that all men are created equal[.]" The Continental Congress signed the document on July 4, 1776. However, the signatures at this point showed that they wished independence: they could not alone achieve it.

Army Bands
One of the attributes of a well-drilled company of soldiers was its military band. The British and Hessian troops drilled to the beat of drums, which carried the rhythm of the march above the noise of musket fire, and provided a way to communicate on the battlefield. "By 1778 soldiers marched at seventy-five 24″ steps per minute in common time and nearly double that (120 steps per minute) when marching in quick time." The music of the fife (a shrill flute) and drum helped build soldier morale.

If the Rebellion could not have good supplies, it would at least have high morale. With the Middle-brook Order, George Washington directed that every officer must provide military music for his troops. This was despite the limited number of instruments. The bands were used to announce the beginning and end of the day, direct troops in battle, and uplift spirits.

Popular Music in the Revolution
One of the two major songs of the Revolution was the hymn Chester, first published in 1770 in "The New England Psalm Singer" and revised in 1778. Its author and composer, William Billings, created a combination of the biblical (Let tyrants shake their iron rod) and the topical (Howe and Burgoyne and Clinton too,/ With Prescot and Cornwallis join'd).

Another was the song Yankee Doodle, adapted from a tune of the Seven Years War. This was originally used by the British to laugh at the provincial manners of the Colonists, but was turned into a theme of the American upstarts.

Canada
In September of 1775, the Colonists, led by General Richard Montgomery, invaded Canada. At first the invasion proved successful, with Montgomery capturing Fort St. Jean and the city of Montreal. On December 30 he made the decision to launch an attack onto the British held city of Quebec. It proved disastrous, and Montgomery was killed in battle. This was the last major action in Canada, although Benidict Arnold and a number of other generals did attack the coasts or Canada, or launch raids across the border.

The Turning Point of the War
Despite the numerous defeats they faced in the early years of the war, the colonists were able to turn the tide around with several major victories.

New York and New Jersey
In July, 1776, General William Howe and thirty-thousand British troops arrived at Staten Island in New York. The large army attacked and defeated General George Washington's American forces in the Battle of Long Island. After nearly having his entire army captured, Washington led a skilled withdrawal out of New York. Eventually the Continental Army was forced to set up camp in Pennsylvania.

Howe could have ended the war by pursuing Washington's forces. But Howe was very cautious and took almost no risks. He feared losing too many men so far from home. Britain hired German mercenaries (Hessians) to guard the British fort at Trenton. Howe took advantage of these replacements and decided to wait until spring to attack the Continental Army again.

Washington also took advantage of the situation, though from a different perspective. He figured that the Hessians would be weakest on Christmas night, after heavy feasting and drinking. On the night of December 25, 1776, Washington led his troops 9 miles, and across the Delaware River to ambush the Hessians. Crossing the river was difficult. A hail and sleet storm had broken out early in the crossing, winds were strong and the river was full of ice floes. The crossing took 3 hours longer than expected, but Washington decided to continue the attack anyway. As Washington predicted, the mercenaries were completely caught off guard and had little time to respond. Within just a over an hour, on the morning of December 26, the Continental Army had won the Battle of Trenton. The Americans had just 4 wounded and 0 killed against 25 Hessians Killed, 90 wounded and 920 captured. The victory increased the troops' morale and eventually led to re-enlistments. Some historians even speculate Trenton saved the revolution.

On January 2, the British came to re-take Trenton, and did so with heavy casualties. Washington once again led a clever withdrawal, and advanced on Princeton. At the Battle of Princeton, the Continental Army attacked the rear-guard of the British Army, and forced them to retreat from New Jersey.

During the war, the New World was being devastated by the 1775–1782 North American smallpox epidemic. Having survived the diseased in his youth, and having been warned about the effect the disease may have on the Army by Benjamin Franklin , George Washington wrote he had more dread of the disease crippling the continental army then the British Troops. On February 5, George Washington ordered the first mass inoculation of troops, following large disruptions caused by smallpox outbreaks. The policy was unpopular among soldiers, but stopped the mass infections from continuing.

The Battle of Saratoga
In the summer of 1777, British General John Burgoyne and General Howe agreed to attack the colonial Army from two sides and defeat it. Howe marched north, winning the Battles of Brandywine and Germantown and eventually capturing Philadelphia. But Burgoyne was not so fortunate. Delayed by natural traps set up by the Continental Army, his troops slowly marched from Canada to Albany. By September of the year, his forces reached Saratoga, where an enormous American Army attacked the troops. In October, General Burgoyne surrendered all his forces to the Americans. General Howe resigned his post, thwarted despite his victories in Pennsylvania.

The Battle of Saratoga proved to be the major turning point in the war. It persuaded France that America had to overthrow Great Britain, and French aid now was introduced to the colonists. The battle was also the last time the British would advance North. By the summer of 1778, following the Battle of Monmouth in New Jersey, all fighting would take place in the South.

Defeat of the Iroquois
The Iroquois Confederacy in its zenith had been the equal of the European Powers. But since the French and Indian war it had been in decline. The Tribes of the Confederacy disagreed on who to support in the Revolution. The Onedia and Tuscaroras supported the Americans, while the Mohawk, Onondaga, Cayuga, and the Seneca supported the British. The Confederacy managed to stay together until 1777, when following the Battle of Saratoga, the 4 Tribes supporting the British began to attack American settlements across New York and Pennslyvenia.

A back and forth battle followed. The Iroquois would attack American Forts and Towns, then the Americans would burn Iroquois villages. In 1779 George Washington sent General Sullivan to destroy the Iroquois Nation. After defeating the Iroquois at the Battle of Newtown, Sullivan's army then carried out a scorched earth campaign, methodically destroying at least forty Iroquois villages. The devastation created great hardships for the thousands of Iroquois refugees outside Fort Niagara that winter, and many starved or froze to death. The survivors fled to British regions in Canada and the Niagara Falls and Buffalo areas. Thus ended the 700-year history of the Iroquois Confederacy.

Conclusion of the War (1778 - 1781)
After the loss at Saratoga, the French, traditional rivals of the British, offered their aid in the Revolution. The United States allied itself with France in 1778. Spain and the Dutch Republic also joined the American side, both lending money to the United States and going to war with Britain.

Valley Forge
Following the capture of Philadelphia by the British, Washington took his followers to Valley Forge on December 17th, 1777, a defensible nearby area, and built camp for 12,000 soldiers and 400 civilians, then the fourth largest settlement in the colonies. Following the introduction of the Prussian protege of Frederick the Great, Baron Von Steuben to Congress by way of Benjamin Franklin, he was directed to Valley Forge, where he arrived on February 23, 1778.

On the Seas
War broke out on the seas as well. Americans granted commissions to "privateers" to attack and destroy all British ships, whether they were military or not. One of the most famous privateers, John Paul Jones, scored several victories at sea for the Americans, even attacking the shores of Britain itself.

The War Heads South
An attempted treachery was defeated when its architect, British Major John Andre, was captured in September of 1780. Benedict Arnold, one of the heroes of Fort Ticonderoga, had been placed in charge of Fort Clinton, New York (now called West Point). In response to a bribe, Arnold neglected maintenance of the fortification, and was then preparing to turn the fort over to the British. After he had learned of Andre's arrest he fled to join the British army.

Britain turned its attention from the North to the South, where more loyalists lived. They were at first very successful, defeating the Americans at Waxhaws, Charleston, and Camden. Lord Cornwallis, commander of the British forces in the south, was faced with the challenge of chasing down the Americans. Nathanael Greene had split his army into two, leaving one under the control of Daniel Morgan. Morgan drew Banastre Tarleton, who was commanding one half of the British Army, to Cowpens where they were they decisively defeated the British. The other half of the British Army, still under control of Cornwallis, defeated the Americans at the Battle of Guilford Court House. However, it was a bloody victory for Cornwallis and he was forced to withdraw to Yorktown Virginia to regroup.

After hearing that the British were in Yorktown, and there was a French Fleet arriving, Washington took the Continental Army, along with French Troops, to Yorktown and surrounded the British. By mid September the town was under siege. Cornwallis was assured by British Commander-in-Chief, Henry Clinton, who was in New York, that he would be relieved shortly. However, the British relief force was defeated by the French fleet. The British continued to hold off for a few more days, but the allied army moved in closer and closer to Yorktown, and their cannons destroyed many of the British defenses. On October 19, 1781, Cornwallis surrendered his entire army, over 7,000 men.

Scattered fighting continued, but back in Britain, the British were crushed by this defeat. Parliament voted to cease all offensive operations in "the colonies." Washington took his army to Newburgh, New York, where he stopped a mutiny in the Army.

At the conclusion of the war in 1783 large numbers of loyalists and their families relocated to the home country of England and in large part to Canada as well as to other British Colonies. They submitted claims for lost property and lands in America. Many of the claims were not accepted by the English government for lack of evidence of the losses or significantly reduced. The property and lands were acquired by the American communities and then resold to the highest bidders.

Due to the climatic effects of a 1782 eruption of an Icelandic volcano, the loyalists also experienced one of the coldest Canadian winters on record which contributed to poor crops in 1783-1784. Starvation, disease and hardship were rampant and many resolved to return to the United States despite the threats of retribution rather than subsist on their meager produce.

Treaty of Paris (1783)
The British lost hope of crushing the rebellion after Yorktown. They decided to negotiate peace with The United States, France, and Spain. The Treaty of Paris was signed on September 3rd, 1783. In it, the United States was recognized as an independent nation, with boundaries stretching from the Canadian border in the North, to the Mississippi River to the West, and the northern border of Florida in the South. Britain was forced to return Florida to Spain, but could still hold Canada. Congress was told to advise the states to restore property lost or stolen from the Loyalists. (However, many Loyalists had fled during the Revolution, and many of them did not return to claim their property.)

Catholics in the Revolution
The complex situation of Catholicism in Great Britain had results in its Colonies. At the time of the American revolution, Catholics formed approximately 1.6% of the total American population of the original 13 colonies. If Catholics were seen as potential enemies of the British state, Irish Catholics, subject to British rule, were doubly-damned. In Ireland they had been subject to British domination. In America Catholics were still forbidden from settling in some of the colonies. Although the head of their faith dwelt in Rome, they were under the official representation of the Catholic Bishop of the London diocese, one James Talbot. When War began, Bishop Talbot declared his faithfulness to the British Crown. (If he had done otherwise, Catholics in England would have been in trouble. Anti-Catholic sentiment still ran high.) He forbade any Colonial priest to serve Communion. This made practice of the faith impossible. This created sympathy for the Colonial rebels. The Continental Army's alliance with the French increased sympathy for the faith. When the French fleet arrived in Newport, Rhode Island, the colony repealed the Act of 1664 and allowed citizenship to Catholics. (This anticipated the provision of the Constitutional Bill of Rights which would strike anti-Catholic laws from the books.) After the war, the Pope created an American Bishop, John Carroll -- a descendant of the same Carrolls who had helped found Maryland -- and an American Diocese communicating directly with Rome.

From Anglicanism to Episcopalianism
On the one hand the colonial Church of England was an organ of the British government and a collaborator with it. Its clergy swore allegiance to the King. Several colonial governments paid monies to the local Anglican Church. Although other faiths were allowed in those states, the Anglican was considered the Official (Established) Church, putting pressure on other denominations. Still, several Revolutionaries, including Thomas Jefferson, rented their pews in a Church of England building. (Jefferson's own faith was Low Church, and he disagreed with the miracles in Christianity.) Significant meetings of the rebellion were held in Church of England buildings.

But after the war, the Church needed to find a new role. Some of the Loyalist clergy went north to Canada. Others were allowed to remain after swearing an oath to the new government. The formerly Established Church was no more: even before the creation of the Constitution, with its separation of Church and State, Americans did not want to pay any extra fees. The Book of Common Prayer, the form of worship in that Church, was pragmatically revised for the new Episcopal Church so that people prayed for "Civil Rulers," instead of the King. But many Church buildings were closed, and there was now room for other denominations to flourish in Virginia and other states.

The Early Government of the New United States
[Copied from Wikipedia]

The Articles of Confederation, formally the Articles of Confederation and Perpetual Union, was an agreement among the 13 founding states that established the United States of America as a confederation of sovereign states and served as its first constitution. Its drafting by the Continental Congress began in mid-1776, and an approved version was sent to the states for ratification in late 1777. The formal ratification by all 13 states was completed in early 1781. Even when not yet ratified, the Articles provided domestic and international legitimacy for the Continental Congress to direct the American Revolutionary War, conduct diplomacy with Europe and deal with territorial issues and Native American relations. Nevertheless, the weakness of the government created by the Articles became a matter of concern for key nationalists. [Whom?]

Questions for Review
1. Who were these authors/composers, and what were they known for? (Mary Otis Warren, Thomas Jefferson, Thomas Paine, William Billings.)

2. How do the battles of Lexington and Concord show the early strengths and weaknesses of the American fighters?

3. Examine a copy of the Declaration of Independence in relation to this and the previous chapters. How does its rhetoric (choice of words) address the concerns of the American rebellion? How does it deviate from actual events to make a point?

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: US History/American Revolution


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US History/Friction Between States

Ideas and Questions of the Time
The overriding question throughout the decade preceding the Civil War was, “Should slavery be allowed in the new territories of the United States?” Before 1848, the question had been hypothetical; however, with the new lands acquired during the Mexican War, it was time for America to make a firm decision regarding the expansion of slavery.

The central ideas dominating the debate were:

The Wilmot Proviso
On August 8, 1846, Representative David Wilmot, a Pennsylvania Democrat, presented a proposal expressing that “slavery nor involuntary servitude shall ever exist in any part of any territory obtained from Mexico.” The Wilmot Proviso was never accepted as law, but it at long last put the issue forth on the political table.

The Calhoun Resolutions
John C. Calhoun, the South Carolina statesman, responded with the Calhoun Resolutions, which said that Congress had no right to stop any citizen with slaves in their possession from taking those slaves into one of the territories. If they did so, the Fifth Amendment, which states that no person can be “deprived of life, liberty, or property, without due process of law,” would be violated. While this was not made formal legislation either, this belief became the standard in most of the south.

Popular Sovereignty
A third option, which appealed to many moderates, most prominently Stephen A. Douglas of Illinois, was the idea of popular sovereignty. This was the idea of letting the settlers of a territory themselves decide whether slavery was to be allowed in it, by voting on state constitutions and other such measures. The primary merit of this initiative was that it took the debate out of Congress, which quickly grew tired of the issue, and put it into the hands of people it truly affected. There was also an unspoken understanding that most of the territories would end up being free, as most settlers that were already in those areas did not bring their slaves with them.

Compromise of 1850
America looked to the Senate for an answer to the question of slavery within the territories. Henry Clay, nicknamed the "Great Compromiser," constructed a compromise: California was admitted as a free state, but all other territories in the Mexican Cession were allowed to choose between becoming a free territory or a slave territory. Also, as part of the Compromise, the slave trade was banned in the District of Columbia, and a Fugitive Slave Act was passed to allow the capture of fugitive slaves.

The Fugitive Slave Act was a very controversial measure. Previously, many in the North felt that slavery merely occurred in the South and that they had nothing to do with it. But under the Fugitive Slave Act, Northerners were required to help return runaway slaves. Thus, the Northerners felt that they were being dragged into aiding the institution of slavery. Several Northern states passed laws prohibiting their officials from aiding the enforcement of the Act.

While the admission of California as a free state gave the free states the majority in Congress, the pro-slavery measures in the Fugitive Slave Act made the Compromise seem more favorable to the South.

Uncle Tom’s Cabin
Harriet Beecher Stowe’s Uncle Tom’s Cabin, published in 1852, is often called “the book that started the Civil War.” The melodramatic story of the evil overseer Simon Legree and his slaves Eliza and Uncle Tom painted an accurate picture of the horrors of slavery, and gave rise to much abolitionist feeling in the North. However, the effects were not easily visible from the start: because the country was growing tired of the sectional bickering over slavery, it took a while for the story to becoming embedded in the American imagination.

Nat Turner
Nat, commonly called Nat Turner, (October 2, 1800 – November 11, 1831) was an American slave whose slave rebellion in Southampton County, Virginia, was the most remarkable instance of black resistance to enslavement in the antebellum southern United States. His methodical slaughter of white civilians during the uprising makes his legacy controversial, but he is still considered by many to be a heroic figure of black resistance to oppression. At birth he was not given a surname, but was recorded solely by his given name, Nat. In accordance with a common practice, he was often called by the surname of his owner, Samuel Turner.

Election of 1852
In one of the less spectacular elections in American history, Senator Franklin Pierce of the Democratic party defeated General Winfield Scott of the Whig party. The Whigs tried to rely on Scott’s heroics as a general during the Mexican war to get him elected, a strategy that proved unsuccessful. Pierce, of New Hampshire, ended up being largely an ineffective president, trying and failing to please both the North and the South.

The Kansas-Nebraska Act and its Effects
Throughout this time, plans were underway for a transcontinental railroad. A question arose as to what Eastern city should be the main terminus. Senator Stephen Douglas of Illinois hoped to advance his own state’s interests by making Chicago the railroad hub. To do this, he suggested a piece of legislation known as the “Kansas-Nebraska Act,” requiring recognition of two new territories, Kansas and Nebraska, west of Missouri and Iowa, respectively. These territories would both help his railroad and solve the overdue issue of the territories in the remainder of the Louisiana Purchase.

But to get the Kansas-Nebraska Act passed, he would have to get the support of Southerners, who wanted a railroad along a more southern route. For this reason, Douglas included in the Act the provision of popular sovereignty in the territories.

This blatantly violated the Missouri Compromise of 1821, which stated that slavery would be prohibited above the 36º30’ line. Douglas therefore opened himself up to the verbal barrage of protests from the North, who denounced the cancellation of the Missouri Compromise as unfair. Yet the Act passed, to the indignation of many Northerners, with the support of President Pierce.

The North
Many in the North figured that if the Missouri Compromise was not an unbreakable law, neither was the Fugitive Slave Act, leading to many demonstrations against it. Boston witnessed the most remarkable of these, leading to many New Englanders turning against Pierce for his support of the Kansas-Nebraska Act.

Political Parties
The Whig party essentially buckled under the pressure of the Kansas-Nebraska Act, with the North condemning it and the South supporting it. Whigs from the North joined some Democrats and Free Soilers that united under the general principle of the Wilmot Proviso, eventually calling themselves the Republican Party and offering its first presidential candidate, John C. Fremont in 1856.

Rachel v. Walker
Rachel v. Walker was a lawsuit involving a slave who, in 1834, sued for her freedom from John Walker in the Supreme Court of Missouri, and won. This result was cited in 1856 in the famous Dred Scott v. Sandford case before the Supreme Court of the United States.[1]

Dred Scott
The question of the constitutionality of Congressional Compromises was decided by the Supreme Court in 1856. In "Scott v. Sanford", the Court ruled against a slave, Dred Scott, who had sued to become free. The Court ruled 7-2 that Scott remained a slave, and there were nine written opinions. The Chief Justice of the United States, Roger Taney, decided that blacks were so inferior that they could not be citizens of the United States, and that, consequently, they could not sue for his freedom (a state issue)in diversity in federal court, and therefore the court lacked jurisdiction. Nevetheless (the biggest "nevertheless" in American history) in a supererogatory effort to settle the question of slavery once and for all, the Marylander Taney ruled that the Missouri Compromise (which had banned the expansion of slavery into the territories north of Missouri) among other laws, was unconstitutional because it restricted the Constitutional right to own property. Many felt that Taney had committed a legal error in his decision. First, Taney had ruled that Scott had no right to sue. The case should have ended there. Taney had ruled on the constitutionality of the Missouri Compromise, which had, under Taney's own ruling that Scott had no right to sue, no bearing upon the case. Thus, the outrage against the Dred Scott decision was increased even more.

John Brown’s Raid
John Brown, an extreme abolitionist known for fighting in Bloody Kanasas, came to the federal arsenal at Harper’s Ferry, Virginia for his last fight. He planned to take over the arsenal, give weapons to the slaves that would support him, and make a center of black power in the Appalachian Mountains that would support slave uprisings in the south.

The raid did not go as planned. Brown did take over the arsenal and took a couple of hostages, but ended up being assaulted by Virginia militia and U.S. Marines under the command of Col. Robert E. Lee of the US 2nd Cavalry. He was tried, convicted, and hanged for treason to the State of Virginia.

However, his raid left a profound impact. John Brown became a martyr for the abolitionist cause during the Civil War. In the South, his actions gave cause to rumors of Northern conspiracy supporting slave insurrections, engendering further suspicion of outsiders in the South. A later Northern marching song sang “John Brown’s body lies a-mouldering in the grave, but his soul is marching on.”

Lincoln
 Lincoln campaign poster In 1860, four major candidates ran for President. The Whigs, adopting the name "Constitutional Union", nominated Tennessean Senator John Bell. The Northern Democrats nominated Senator Stephen Douglas of Illinois and the Southern Democrats nominated the Vice President John Breckenridge of Kentucky. The more united Republican party nominated Abraham Lincoln, who spoke out against expansion of slavery. Though he assumed that, under the constitution, Congress could not outlaw slavery in the South, he assured all that he would work to admit only free states to the US. Due to divisions between the parties, Lincoln won the election by carrying every Northern State. Douglas won Missouri, Bell the Upper South, and Breckenridge the Deep South. The South was outraged. The North had a far larger population than the South, and thus had more electoral votes. The South had been out voted.

title: US History/Friction Between States


text:
US History/Preface

This textbook is based on the College Entrance Examination Board test in Advanced Placement United States History. The test is a standard on the subject, covering what most students in the United States study in high school and college, so we treat it as the best reference.

The text was reorganized and edited in November 2008 to be closer to the content and organization the college board requires. The content was carefully chosen for significance and interest. We welcome reader feedback and suggestions for improvement.

Enjoy! The AP Course Description can be found here.

title: US History/Preface


text:
GCSE Science/Circuits Part2

GCSE Science/Electricity

Now we have learned Ohm's Law we can start applying it to some circuits.

The test circuits below can be used to investigate how the voltage across a component varies with the current flowing through the component. Either circuit can be used. The one on the left is easiest to set up, the one on the right is easiest to use once it is set up, but does require you to use the rheostat as a potential divider.

Note that the voltmeter is in parallel with the component and the ammeter is in series with it.This is necessary so that the ammeter and voltmeter do not interfere with the circuit in anyway. An ideal meter does not change either the current or the voltage.

The wire should be resistance wire, such as nichrome. Ordinary copper wire would short circuit the power pack and blow the pack's fuse.

Set up either circuit and then by turning the voltage setting on the power pack and the slider on the rheostat, adjust the current to read 0.1 A. Take a reading of the voltage. Repeat for 0.2 A, 0.3 A up to 1.0 A. Repeat the readings but this time go from 1.0 A to 0.9 A and so on down to 0.1 A.

Set your results out in a table like this.

Now plot the average voltage against the current on graph paper. You should find the points all fall approximately on a straight line. The slope of the line =Voltage/Current. This is the resistance.

Below is a typical voltage/current graph for a wire. Note that all the points fall approximately on a straight line.

The slope of the line is 1.1 V/A.

Therefore the resistance of this wire is 1.1 Ω

All Ohmic components have a constant resistance like this. However, non-Ohmic components (that is, components that do not obey Ohm's law) do not have such a resistance. A bulb, for example, has a resistance that increases as the current flowing through it goes up, because the filament is heating up. A current-voltage graph for a bulb would be a curve.

Q1) Plot a current/voltage graph for the following set of data for a 12V filament bulb.

Answers | <>

title: GCSE Science/Circuits Part2


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Algebra/Complex Numbers

 * See also: High School Mathematics Extensions, Complex numbers, Complex numbers at Wikipedia

Complex numbers are the extension of the real numbers, i.e., the number line, into a number plane. They allow us to turn the rules of plane geometry into arithmetic. Complex numbers have fundamental importance in describing the laws of the universe at the subatomic level, including the propagation of light and quantum mechanics. They also have practical uses in many fields, including signal processing and electrical engineering.

Introduction
Currently, we are able to solve many different kinds of equations for $$x$$, such as $$x+7=12$$ , or $$3x=4$$ , or $$x+10=0$$. In each of these cases the solution for $$x$$ is a real number: respectively 5, 4/3, and -10.

However, there is no real number x that satisfies the equation $$x^2=-1$$, since the square of any real number is nonnegative.

Conceptually it would be nice to have some kind of number to be the solution of $$x^2=-1$$. This "number" would not be a real number, however, and we refer to such a number as an imaginary number.

Now, is this really the reason?

Well - Definitely not!

This mistake occurs in many teaching books from the attempts to "solve problems by force", as we could explain psychologically. This has nothing to do with reality, and gives the false feeling that mathematicians are "Cranks" full of to much spare time on their hands with nothing to do.

The reason, be surprised, has to do with a problem called Cubic functions.

We then extend the real number system to accommodate this special number. It turns out that there will be two imaginary solutions of the equation $$x^2=-1$$. One of them will be called $$i$$ and, following the normal rules for arithmetic, the other solution is $$-i$$. We may be inclined to say that $$\sqrt{-1}=i$$. That would, however, be incorrect solely because in words this says that "the square root of -1 is $$i$$", but there is no basis for preferring $$i$$ over $$-i$$ (or vice versa) as the square root of -1. Rather, the two square roots have equal standing.

We say that all numbers of the form a +b$$i$$, where $$a$$ and $$b$$ are any real numbers, is the set of complex numbers, and we denote this set $$\Complex$$. The real numbers $$\R$$ may be considered to be the subset of complex numbers $$\Complex=\{a+bi\}$$ for which b = 0. Complex numbers can be added, subtracted, multiplied, and divided (except by 0). We will explore some of the properties of these numbers later.

There are in fact two commonly used definitions of complex numbers, but they are immediately seen to be logically equivalent.

For a negative root like $$\sqrt{-4}$$, we split the number into two parts such that one part is $$\sqrt{-1}$$ like $$\sqrt{-4}=\sqrt{4(-1)}=\sqrt{4}\cdot\sqrt{-1}$$ which leads to $$2i$$

Definition 1
A complex number is an expression of the form x + yi, in which x and y are real numbers and i is a new number, called the imaginary unit, for which expressions the normal rules of calculation apply together with the extra rule: i2=-1.

Definition 2
A complex number is a pair of real numbers (x,y), satisfying the properties:

 * $$(a,b)+(c,d)=(a+c,b+d)$$
 * $$(a,b)\cdot(c,d)=(ac-bd,ad+bc)$$

In both cases a complex number consists of two real numbers x and y. The real number x is called the real part and the real number y the imaginary part of the complex number.

From the properties we deduce that complex numbers of the form (x,0) behave just like the real numbers, so we identify (1,0) with 1 and hence (x,0) with x. Furthermore we see that:

 * $$(0,1)\cdot(0,1)=(-1,0)=-1$$.

It is common use to write i instead of (0,1), so:
 * $$i=(0,1)$$ and $$i^2=-1$$.

Any complex number (x,y) may now be written as x + yi.

Some examples
A complex number is a number that is in the form $$z=a+bi$$, where a and b are real numbers. We say that a is the real part of z and write $${\rm Re}(z)$$, and that b is the imaginary part of z, and write $${\rm Im}(z)$$

A number of the form bi is sometimes called a pure imaginary number, as it has no real part. The pure imaginary numbers are also complex numbers, because bi = 0 + bi. In the same way, all real numbers are also complex numbers, because a = a + 0i. So the set of complex numbers includes real numbers, pure imaginary numbers, and the sums of reals and pure imaginaries.

Here are some examples
 * 1 + 4i: a complex number, real part 1, imaginary part 4
 * 2 - 2i: a complex number, real part 2, imaginary part -2.
 * -4i: a complex number, real part 0, imaginary part -4
 * 2: a complex number (also a real number), real part 2, imaginary part 0.

Notice that the number 2 is a complex number and a real number. This fact is clearer if we write 2 = 2 + 0i.

Any complex number may be written in three main forms, which we will explore later. The form x + yi is known as the Cartesian form.

Complex numbers and matrices
Complex numbers can be identified with a certain set of matrices. If we think of the 2×2 identity matrix as the number 1, and we think of $$i$$ which we introduced above as the matrix
 * $$\begin{pmatrix}

0 & -1 \\ 1 & 0 \end{pmatrix}$$,

then the complex number $$a+bi$$ then has the form
 * $$\begin{pmatrix}

a & -b \\ b & a \end{pmatrix}$$.

Properties
Complex numbers obey most of the properties of real numbers. Take two complex numbers, $$z_1=a+bi\,$$ and $$z_2=c+di\,$$.

Addition
How can we add these two complex numbers?

We don't even have to think about $$i$$ as being "special" in any way, just treat it as any other symbol and proceed by the standard rules of algebra, grouping along the way.

We obtain:
 * $$z_1+z_2=(a+bi)+(c+di)=(a+c)+(b+d)i\,$$

If one uses the matrix analogy above, regular matrix addition works to add complex numbers in the same way. Verify for yourself that this is true.

Subtraction
Subtraction proceeds just as before.
 * $$z_1-z_2=(a+bi)-(c+di)=(a-c)+(b-d)i\,$$

Multiplication
By the normal rules, taking into account that $$i^2=-1\,$$, we find:
 * $$z_1z_2=(a+bi)(c+di)=ac+adi+bci+bdi^2=(ac-bd)+(bc+ad)i\,$$

If one uses the matrix analogy above, regular matrix multiplication works to multiply complex numbers in the same way. Verify for yourself that this is true.

Conjugates
The conjugate of a complex number $$z$$, written $$\bar{z}$$, is the same number with the sign of the imaginary part changed: the conjugate of a + bi = a - bi (and vice versa).

Let us examine what happens when we have a complex number z = a + bi, what is the product of z and its conjugate?

 * (a+bi)(a-bi) = a2 + abi - abi - b2i2 = a2  +  b2

Notice the imaginary parts cancel out, so the product is a real number. This will aid us greatly in the division of a complex number, as we will see.

Notice also that this is the sum of two squares, analogous to the difference of two squares.

Matrix transposition behaves as conjugation if one uses the matrix analogy.

Division
How do we compute the quotient
 * $$\frac{a+bi}{c+di}$$

of two complex numbers? It is not difficult. Let the quotient be:

 * $$\frac{a+bi}{c+di}=r+si$$,

then cross multiplication gives:
 * $$\,a+bi=(r+si)(c+di)=rc-sd+i(rd+sc)$$,

hence
 * $$\,a=rc-sd$$,

and
 * $$\,b=rd+sc$$.

So we have to solve two linear equations. Solution:

 * $$r=\frac{ac+bd}{c^2+d^2}$$,

and
 * $$s=\frac{bc-ad}{c^2+d^2}$$.

Note that this is complete nonsense unless $$c^2+d^2 \ne 0$$. In fact, it is easy to see that the pair of linear equations have a solution exactly when this is true, i.e., when c and d are not both zero. Or in other words, when $$c+di \ne 0$$ as a complex number. Thus we can divide by a complex number only when it is non-zero.

Luckily there is a little trick to speed up this computation. We multiply the denominator with a well chosen number as to make it real. We realize the denominator by multiplying with the conjugate of the denominator; a complex number times its conjugate is a real number:

 * $$\,(c+di)(c-di)=c^2-(di)^2=c^2+d^2$$.

hence:
 * $$\frac{a+bi}{c+di} = \frac{a+bi}{c+di}\cdot \frac{c-di}{c-di} = \frac{ac-adi+bci+bd}{c^2+d^2} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}=\frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i$$.

Note that in the multiplication and division of complex numbers, we usually work out the whole problem instead of just memorizing the equation of the answer. The reader will note that we use here the familiar trick from algebra of multiplying by the number 1 in a particularly convenient form: $$\frac{c-di}{c-di}$$. (We leave it to the reader to verify that any non-zero complex number divided by itself is in fact equal to 1.)

Exponents and Roots
Because $$x^{y}=e^{ln{x}\times y}=1+\frac{ln{ x}\times y}{1!}+\frac{(ln{ x}\times y)^{2}}{2!}+\frac{(ln{ x}\times y)^{3}}{3!}+...$$, real numbers can be raised to imaginary numbers. Imaginary and complex numbers cannot be raised to imaginary or complex numbers, as imaginary and complex numbers have no natural logarithms. Example: $$e^{i}=0.5403023058681397+0.8414709848078965i$$ Because $$\sqrt[y]{x}=x^{\frac{1}{y}}$$, imaginary numbers can be root degrees.

Problem set
Given the above rules, answer the following questions.

Note: Use sqrt(x) for $$\sqrt{x}$$  { (7 + 2i) + (11 - 6i) = { 18_19 } + { -4_19 } i
 * type="{}"}

{ (8 - 3i) - (6i) = { 8_19 } + { -9_19 } i
 * type="{}"}

{ (9 + 4i)(3 - 16i) = { 91_19 } + { -132_19 } i
 * type="{}"}

{ 3i $$\times$$ 9i = { -27_19 } + { 0_19 } i
 * type="{}"}

{ $$\frac{i}{2+i}=$${ 1/5|0.2_19 } + { 2/5|0.4_19 } i
 * type="{}"}

{ $$\frac{11 + 3i}{\sqrt{3} - 4i}=$${ [11sqrt(3) - 12]/19 (i)|[11sqrt(3)-12]/19 (i) _19 } + { [44 + 3sqrt(3)]/19 (i)|[44+3sqrt(3)]/19 (i) _19 } i
 * type="{}"}

{ $${(x + yi)}^{-1}=$${ x/(x^2 + y^2) (i)|x/(x^2+y^2) (i) _19 } + { -y(x^2 + y^2) (i)|-y(x^2+y^2) (i) _19 } i
 * type="{}"}

The Argand Plane
We can represent complex numbers geometrically as well. Every complex number can be represented in the form z=x+iy (so x=Re(z) and y=Im(z)). Then we can represent z in the xy-plane by the point (x,y). Notice that this is a one-to-one relationship: for each complex number, we have one corresponding point in the plane, and for each point in the plane there corresponds one complex number. When we use the xy-plane in this way to represent complex numbers, we call the plane the "Argand plane". We will refer to the "y" axis as the imaginary axis, and the "x" axis as the real axis.

Notice that a purely imaginary number is represented in the Argand plane by a point on the imaginary axis. A purely real number is represented by a point on the real axis.

Here is an example of the Argand plane.
 * [[Image:Argand2.png]]

There are two complex numbers drawn in the plane; viz., 1 + i, -2 - i. Their sum is plotted on the graph, -1. The red and blue lines demonstrate how geometrically, a parallelogram can be constructed, and their apex forms their sum.

Modulus and argument

 * [[Image:Modulus and argument.png]]

On this diagram we can see the number 3 + 4i. The red line is the distance away from the origin (the number 0 + 0i). The gray line represents the distances away from the respective axes. We can see that the red line makes an angle θ from the real axis.

It is clear that almost all complex numbers have this distance away from the origin and that almost all complex numbers make an angle away from the real axis. We give these two qualities special names; the distance away from the origin is known as the modulus of the complex number, and the angle θ is known as the argument of the complex number.

We write the modulus of a complex number z by |z|, and the argument of the complex number as arg z.

We can calculate the modulus and argument by basic trigonometry.

Calculating the modulus
In the above example, we have the number 3 + 4i. We can form a triangle in the Argand plane with base 3 and height 4. By Pythagoras we can find the length of the hypotenuse by $$\sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5$$.

And thus, the length of the hypotenuse is thus the modulus of the complex number, and it is 5 for 3+4i.

Generalization
If z = x + yi, |z| is clearly $$\sqrt{x^2+y^2}$$.

Equivalently, $$|z|=\sqrt{\mathrm{Re}(z)^2+\mathrm{Im}(z)^2}$$.

Calculating the argument
We have the same triangle as we had in calculating the modulus. Remember from trigonometry that tan θ is the ratio of the height over the base. So, for 3 + 4i, we have tan θ = 4/3, and thus θ = arctan 4/3 = 0.9...

With complex numbers, we always take two things:
 * the argument must be in radians
 * the argument lies in the interval [-π,π], and we always adjust the angle so it does.

Note that arg 0 is undefined.

Generalization
If z = x + yi, arg z is clearly arctan (y/x), or, equivalently, arg z = arctan (Im(z)/Re(z)).

The polar form
We are now able to calculate the modulus and argument of a complex number, where these two numbers are able to uniquely describe every number in the Argand plane.

Using these two characteristics of complex numbers, we are now able to formulate a new way of writing these numbers.

Note that in the above diagram, we obtain a triangle that describes the complex number 3 + 4i. Clearly, we can do this for all complex numbers in the Argand plane (except for 0).

To simplify our work, let us look at numbers in the circle of unit length equidistant from 0. From trigonometry, we can parameterize all the points on a circle in the Cartesian plane by (cos θ, sin θ). In complex number notation we can say that all numbers on this unit circle are in the form cos θ+i sin θ.

This works well on the unit circle, but how does this generalize to describing all numbers on the plane? We simply make the circle larger or smaller to encompass the number; this is done by multiplying by the modulus.

So then, we obtain the polar form r(cos θ + i sin θ) = z, where r is the modulus.

Euler's formula
A very significant result in the area of complex numbers is Euler's formula. It basically asserts that
 * reiθ = r (cos θ + i sin θ)

This statement can be verified through a rearrangement of the Taylor Series of the cosine and sine functions.

Note that conjugate complex numbers have an opposing argument. 2e2i and 2e-2i are conjugate pairs.

Using Taylor series
Here is a proof of Euler's formula using Taylor Series expansions as well as basic facts about the powers of i:

i^0 = 1, \ i^1 = i, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1, \ i^5 = i, \ i^6 = -1, \ i^7 = -i, \ i^8 = 1, \dots $$

The functions ex, cos(x) and sin(x) (assuming x is real) can be written as:

 * $$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $$

 * $$ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

$$

 * $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

$$

and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that

 * $$e^{iz} = 1 + iz + \frac{(iz)^2}{2!} + \frac{(iz)^3}{3!} + \frac{(iz)^4}{4!} + \frac{(iz)^5}{5!} + \frac{(iz)^6}{6!} + \frac{(iz)^7}{7!} + \frac{(iz)^8}{8!} + \cdots$$

 * $$= 1 + iz - \frac{z^2}{2!} - \frac{iz^3}{3!} + \frac{z^4}{4!} + \frac{iz^5}{5!} - \frac{z^6}{6!} - \frac{iz^7}{7!} + \frac{z^8}{8!} + \cdots$$

 * $$= \left( 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} + \cdots \right) + i\left( z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots \right) $$

 * $$= \cos (z) + i\sin (z) \,$$

The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number, gives the original identity as Euler discovered it.$$\square$$

Using calculus
Define the complex number z such that
 * $$z=\cos x + i\sin x \,$$

Differentiating z with respect to x:
 * $$\frac{dz}{dx}=-\sin x + i\cos x$$

Using the fact that i2 = -1:
 * $$\frac{dz}{dx}=i^2\sin x + i\cos x=i(\cos x + i\sin x)=zi$$

Separating variables and integrating both sides:
 * $$\int\frac{1}{z}\,dz=\int i\,dx$$
 * $$\ln z=xi + C\,$$

where C is the constant of integration. To finish the proof we have to argue that it is zero. This is easily done by substituting x = 0.
 * $$\ln z = C\,$$

But z is just equal to:
 * $$\ z = \cos x + i\sin x = \cos 0 + i \sin 0 = 1\,$$

thus
 * $$\ln 1 = C\,$$

 * $$\ C = 0\,$$

So now we just exponentiate
 * $$\ \ln z = xi\,$$

 * $$\ e^{\ln z} = e^{xi}\,$$

 * $$\ z = e^{xi}\,$$

 * $$\ e^{xi} = \cos x + i\sin x\,\square$$

Corollaries
A number of significant results follow as corollaries to Euler's result.

de Moivre's theorem
De Moivre's theorem is useful in calculating powers of complex numbers. It states that
 * (r (cos θ + i sin θ))n = rn ( cos nθ + i sin nθ)

This follows clearly (from the laws of exponents) if we rewrite the theorem in the form
 * (reiθ)n=rneinθ

Equivalent trigonometric forms
From the cosine/sine form of the complex number, we can rewrite the cosine and sine function in terms of exponentials.
 * cos θ = 1/2 (eiθ + e-iθ)
 * sin θ = 1/(2i) (eiθ - e-iθ)

Relation of e, $$\pi$$, i, 1, and 0
By substituting π into the formula, we obtain the following result:

$$e^{\pi i}+1=0$$

The actual mathematical relevance of this equation is actually very little. It is known more for its relation of the many branches of mathematics: e comes from calculus, π from geometry, i comes from algebra, 1 is the multiplicative identity, and 0 is the additive identity. It is also known for its simple mathematical aesthetic.

Forming trigonometric identities
The aforementioned equivalent trigonometric forms, combined with the Binomial theorem, allow us to create some trigonometric identities that would be difficult to form in any other way. These identities can be used to simplify integral problems.

Cosine/sine powers
How can we simplify, say, (cos x)5?

Let us look at a simple example for motivation. First, rewrite as
 * (cos x)5 = (1/2 (eix+e-ix))5 =

1/25 (eix+e-ix)5

By the Binomial theorem,
 * (a+b)5=a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.

Replacing a with eix/2 and b with e-ix/2, we obtain
 * 10e-ix/32+10eix/32+5e3ix/32+5e-3ix/32+e5ix/32+e-5ix/32=
 * (5/16)(e-ix+eix)+(5/32)(e3ix+e-3ix)+(1/32)(e5ix+e-5ix)=
 * (5/16)(2 cos x)+(5/32)(2 cos 3x)+(1/32)(2 cos 5x)=
 * (5/8) cos x+(5/16) cos 3x+(1/16) cos 5x=(cos x)5

Procedure
We can summarize from the above example the general procedure: To simplify an expression in the form:
 * cos(x)k
 * sin(x)k

the procedure is:
 * write cos(x) or sin(x) in the exponential form, all to the power of k
 * expand using the Binomial theorem
 * collect conjugate pairs
 * write back from the exponential form into the trigonometric form

Cosine/sine multiples
We can also form identities in the form:
 * cos(kx)
 * sin(kx).

Let's look at another example to see how it's done.

Example
Let's expand sin(3x).

Recall de Moivre's theorem stating
 * (cos(x)+i sin(x))3 = cos(3x)+i sin (3x)

We will use this fact to expand out the left side. For ease of manipulation, it may be easier to let c = cos(x), and s=sin(x). Then use the Binomial theorem again to expand out:
 * (c+is)3=c3 + 3i c2s - 3cs2 - i s3

Collect real and imaginary parts
 * (c+is)3=c3 - 3cs2 + i(3c2s - s3)

Now, this is of course equal to cos(3x)+i sin (3x). So, substituting back cos(x) for c and similarly for sin(x), we can equate real and imaginary parts, and we get
 * sin(3x)=3 cos(x)2sin(x) - sin(x)3

and we get
 * cos(3x)=cos(x)3 - 3 cos(x)sin(x)2

for free.

NB: In the cosine expansion, one could write sin(x)2 as 1-cos(x)2 to obtain a formula consisting completely of cosines. (analogously one could write the sine expansion using only sines)

title: Algebra/Complex Numbers