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771_7 | In May, 4,500 British reinforcements arrived under Generals William Howe, John Burgoyne, and Sir Henry Clinton. On June 17, they seized the Charlestown Peninsula at the Battle of Bunker Hill, a frontal assault in which they suffered over 1,000 casualties. Dismayed at the costly attack which had gained them little, Gage appealed to London for a larger army to suppress the revolt, but instead was replaced as commander by Howe.
On June 14, 1775, Congress took control of Patriot forces outside Boston, and Congressional leader John Adams nominated George Washington as commander-in-chief of the new Continental Army. Washington previously commanded Virginia militia regiments in the French and Indian War, and on June 16, John Hancock officially proclaimed him "General and Commander in Chief of the army of the United Colonies." On July 3, He assumed command on July 3, preferring to fortify Dorchester Heights outside Boston rather than assaulting it. In early March 1776, Colonel Henry Knox arrived with heavy artillery acquired in the Capture of Fort Ticonderoga. Under cover of darkness, on March 5 Washington placed these on Dorchester Heights, from where they could fire on the town and British ships in Boston Harbor. Fearing another Bunker Hill, Howe evacuated the city on March 17 without further loss and sailed to Halifax, Nova Scotia, while Washington moved south to New York City.
Beginning in August 1775, American privateers raided towns in Nova Scotia, including Saint John, Charlottetown and Yarmouth. In 1776, John Paul Jones and Jonathan Eddy attacked Canso and Fort Cumberland respectively. British officials in Quebec began negotiating with the Iroquois for their support, while the Americans urged them to maintain neutrality. Aware of Native American leanings toward the British and fearing an Anglo-Indian attack from Canada, Congress authorized an invasion of Quebec in April 1775. A second American invasion was defeated at the Battle of Quebec on December 31, and after a loose siege the Americans withdrew on May 6, 1776. A failed counter-attack at Trois-Rivières on June 8 ended American operations in Quebec. | [
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771_8 | However, British pursuit was blocked by American ships on Lake Champlain until they were cleared on October 11 at the Battle of Valcour Island. The American troops were forced to withdraw to Fort Ticonderoga, ending the campaign. In November 1776, a Massachusetts-sponsored uprising in Nova Scotia during the Battle of Fort Cumberland was dispersed. The cumulative failures cost the Patriots support in local public opinion, and aggressive anti-Loyalist policies in the New England colonies alienated the Canadians. The Patriots made no further attempts to invade north.
In Virginia, an attempt by Governor Lord Dunmore to seize militia stores on April 20 1775 led to an increase in tension, although conflict was avoided for the time being. This changed after the publication of Dunmore's Proclamation on November 7, 1775, promising freedom to any slaves who fled their Patriot masters and agreed to fight for the Crown. British forces were defeated at Great Bridge on December 9 and took refuge on British ships anchored near the port of Norfolk. When the Third Virginia Convention refused to disband its militia or accept martial law, Dunmore ordered the Burning of Norfolk on January 1, 1776.
The siege of Savage's Old Fields began on November 19 in South Carolina between Loyalist and Patriot militias, and the Loyalists were subsequently driven out of the colony in the Snow Campaign. Loyalists were recruited in North Carolina to reassert British rule in the South, but they were decisively defeated in the Battle of Moore's Creek Bridge. A British expedition sent to reconquer South Carolina launched an attack on Charleston in the Battle of Sullivan's Island on June 28, 1776, but it failed and left the South under Patriot control until 1780.
A shortage of gunpowder led Congress to authorize a naval expedition against The Bahamas to secure ordnance stored there. On March 3, 1776, an American squadron landed at Nassau and encountered minimal resistance, confiscating what supplies they could before sailing for home on March 17. A month later, after a brief skirmish with , they returned to New London, Connecticut, the base for American naval operations during the Revolution.
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771_9 | British New York counter-offensive
After regrouping at Halifax, Nova Scotia, William Howe was determined to take the fight to the Americans. He sailed for New York in June 1776 and began landing troops on Staten Island near the entrance to New York Harbor on July 2. The Americans rejected Howe's informal attempt to negotiate peace on July 30; Washington knew that an attack on the city was imminent and realized that he needed advance information to deal with disciplined British regular troops. On August 12, 1776, Patriot Thomas Knowlton was given orders to form an elite group for reconnaissance and secret missions. Knowlton's Rangers, which included Nathan Hale, became the Army's first intelligence unit. When Washington was driven off Long Island he soon realized that he would need more than military might and amateur spies to defeat the British. He was committed to professionalizing military intelligence, and with the aid of Benjamin Tallmadge, they launched the six-man Culper spy ring. The efforts of Washington and the Culper Spy Ring substantially increased effective allocation and deployment of Continental regiments in the field. Over the course of the war Washington spent more than 10 percent of his total military funds on intelligence operations.
Washington split his army into positions on Manhattan Island and across the East River in western Long Island. On August 27 at the Battle of Long Island, Howe outflanked Washington and forced him back to Brooklyn Heights, but he did not attempt to encircle Washington's forces. Through the night of August 28, General Henry Knox bombarded the British. Knowing they were up against overwhelming odds, Washington ordered the assembly of a war council on August 29; all agreed to retreat to Manhattan. Washington quickly had his troops assembled and ferried them across the East River to Manhattan on flat-bottomed freight boats without any losses in men or ordnance, leaving General Thomas Mifflin's regiments as a rearguard. | [
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771_10 | General Howe officially met with a delegation from Congress at the September Staten Island Peace Conference, but it failed to conclude peace as the British delegates only had the authority to offer pardons and could not recognize independence. On September 15, Howe seized control of New York City when the British landed at Kip's Bay and unsuccessfully engaged the Americans at the Battle of Harlem Heights the following day. On October 18 Howe failed to encircle the Americans at the Battle of Pell's Point, and the Americans withdrew. Howe declined to close with Washington's army on October 28 at the Battle of White Plains, and instead attacked a hill that was of no strategic value.
Washington's retreat isolated his remaining forces and the British captured Fort Washington on November 16. The British victory there amounted to Washington's most disastrous defeat with the loss of 3,000 prisoners. The remaining American regiments on Long Island fell back four days later. General Sir Henry Clinton wanted to pursue Washington's disorganized army, but he was first required to commit 6,000 troops to capture Newport, Rhode Island to secure the Loyalist port. General Charles Cornwallis pursued Washington, but Howe ordered him to halt, leaving Washington unmolested.
The outlook was bleak for the American cause: the reduced army had dwindled to fewer than 5,000 men and would be reduced further when enlistments expired at the end of the year. Popular support wavered, morale declined, and Congress abandoned Philadelphia and moved to Baltimore. Loyalist activity surged in the wake of the American defeat, especially in New York state.
In London, news of the victorious Long Island campaign was well received with festivities held in the capital. Public support reached a peak, and King George III awarded the Order of the Bath to Howe. Strategic deficiencies among Patriot forces were evident: Washington divided a numerically weaker army in the face of a stronger one, his inexperienced staff misread the military situation, and American troops fled in the face of enemy fire. The successes led to predictions that the British could win within a year. In the meantime, the British established winter quarters in the New York City area and anticipated renewed campaigning the following spring. | [
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771_11 | Two weeks after Congress withdrew to safer Maryland, Washington crossed the ice-choked Delaware River about 30 miles upriver from Philadelphia on the night of December 25–26, 1776. His approach over frozen trails surprised Hessian Colonel Johann Rall. The Continentals overwhelmed the Hessian garrison at Trenton, New Jersey, and took 900 prisoners. The celebrated victory rescued the American army's flagging morale, gave new hope to the Patriot cause, and dispelled much of the fear of professional Hessian "mercenaries". Cornwallis marched to retake Trenton but was repulsed at the Battle of the Assunpink Creek; in the night of January 2, Washington outmaneuvered Cornwallis and defeated his rearguard in the Battle of Princeton the following day. The two victories helped to convince the French that the Americans were worthy military allies.
Washington entered winter quarters from January to May 1778 at Morristown, New Jersey, and he received the Congressional direction to inoculate all Continental troops against smallpox. Although a Forage War between the armies continued until March, Howe did not attempt to attack the Americans over the winter of 1776–1777.
British northern strategy fails
The 1776 campaign demonstrated regaining New England would be a prolonged affair, which led to a change in British strategy. This involved isolating the north from the rest of the country by taking control of the Hudson River, allowing them to focus on the south where Loyalist support was believed to be substantial. In December 1776, Howe wrote to the Colonial Secretary Lord Germain, proposing a limited offensive against Philadelphia, while a second force moved down the Hudson from Canada. Germain received this on February 23, 1777, followed a few days later by a memorandum from Burgoyne, then in London on leave.
Burgoyne supplied several alternatives, all of which gave him responsibility for the offensive, with Howe remaining on the defensive. The option selected required him to lead the main force south from Montreal down the Hudson Valley, while a detachment under Barry St. Leger moved east from Lake Ontario. The two would meet at Albany, leaving Howe to decide whether to join them. Reasonable in principle, this did not account for the logistical difficulties involved and Burgoyne erroneously assumed Howe would remain on the defensive; Germain's failure to make this clear meant he opted to attack Philadelphia instead. | [
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771_12 | Burgoyne set out on June 14, 1777, with a mixed force of British regulars, German auxiliaries and Canadian militia, and captured Fort Ticonderoga on July 5. As General Horatio Gates retreated, his troops blocked roads, destroyed bridges, dammed streams, and stripped the area of food. This slowed Burgoyne's progress and forced him to send out large foraging expeditions; on one of these, more than 700 British troops were captured at the Battle of Bennington on August 16. St Leger moved east and besieged Fort Stanwix; despite defeating an American relief force at the Battle of Oriskany on August 6, he was abandoned by his Indian allies and withdrew to Quebec on August 22. Now isolated and outnumbered by Gates, Burgoyne continued onto Albany rather than retreating to Fort Ticonderoga, reaching Saratoga on September 13. He asked Clinton for support while constructing defenses around the town.
Morale among his troops rapidly declined, and an unsuccessful attempt to break past Gates at the Battle of Freeman Farms on September 19 resulted in 600 British casualties. When Clinton advised he could not reach them, Burgoyne's subordinates advised retreat; a reconnaissance in force on October 7 was repulsed by Gates at the Battle of Bemis Heights, forcing them back into Saratoga with heavy losses. By October 11, all hope of escape had vanished; persistent rain reduced the camp to a "squalid hell" of mud and starving cattle, supplies were dangerously low and many of the wounded in agony. Burgoyne capitulated on October 17; around 6,222 soldiers, including German forces commanded by General Riedesel, surrendered their arms before being taken to Boston, where they were to be transported to England.
After securing additional supplies, Howe made another attempt on Philadelphia by landing his troops in Chesapeake Bay on August 24. He now compounded failure to support Burgoyne by missing repeated opportunities to destroy his opponent, defeating Washington at the Battle of Brandywine on September 11, then allowing him to withdraw in good order. After dispersing an American detachment at Paoli on September 20, Cornwallis occupied Philadelphia on September 26, with the main force of 9,000 under Howe based just to the north at Germantown. Washington attacked them on October 4, but was repulsed. | [
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771_13 | To prevent Howe's forces in Philadelphia being resupplied by sea, the Patriots erected Fort Mifflin and nearby Fort Mercer on the east and west banks of the Delaware respectively, and placed obstacles in the river south of the city. This was supported by a small flotilla of Continental Navy ships on the Delaware, supplemented by the Pennsylvania State Navy, commanded by John Hazelwood. An attempt by the Royal Navy to take the forts in the October 20 to 22 Battle of Red Bank failed; a second attack captured Fort Mifflin on November 16, while Fort Mercer was abandoned two days later when Cornwallis breached the walls. His supply lines secured, Howe tried to tempt Washington into giving battle, but after inconclusive skirmishing at the Battle of White Marsh from December 5 to 8, he withdrew to Philadelphia for the winter.
On December 19, the Americans followed suit and entered winter quarters at Valley Forge; while Washington's domestic opponents contrasted his lack of battlefield success with Gates' victory at Saratoga, foreign observers such as Frederick the Great were equally impressed with Germantown, which demonstrated resilience and determination. Over the winter, poor conditions, supply problems and low morale resulted in 2,000 deaths, with another 3,000 unfit for duty due to lack of shoes. However, Baron Friedrich Wilhelm von Steuben took the opportunity to introduce Prussian Army drill and infantry tactics to the entire Continental Army; he did this by training "model companies" in each regiment, who then instructed their home units. Despite Valley Forge being only twenty miles away, Howe made no effort to attack their camp, an action some critics argue could have ended the war.
Foreign intervention
Like his predecessors, French foreign minister Vergennes considered the 1763 Peace a national humiliation* and viewed the war as an opportunity to weaken Britain. He initially avoided open conflict, but allowed American ships to take on cargoes in French ports, a technical violation of neutrality. Although public opinion favored the American cause, Finance Minister Turgot argued they did not need French help to gain independence and war was too expensive. Instead, Vergennes persuaded Louis XVI to secretly fund a government front company to purchase munitions for the Patriots, carried in neutral Dutch ships and imported through Sint Eustatius in the Caribbean. | [
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771_14 | Many Americans opposed a French alliance, fearing to "exchange one tyranny for another", but this changed after a series of military setbacks in early 1776. As France had nothing to gain from the colonies reconciling with Britain, Congress had three choices; making peace on British terms, continuing the struggle on their own, or proclaiming independence, guaranteed by France. Although the Declaration of Independence in July 1776 had wide public support, Adams was among those reluctant to pay the price of an alliance with France, and over 20% of Congressmen voted against it. Congress agreed to the treaty with reluctance and as the war moved in their favor increasingly lost interest in it.
Silas Deane was sent to Paris to begin negotiations with Vergennes, whose key objectives were replacing Britain as the United States' primary commercial and military partner while securing the French West Indies from American expansion. These islands were extremely valuable; in 1772, the value of sugar and coffee produced by Saint-Domingue on its own exceeded that of all American exports combined. Talks progressed slowly until October 1777, when British defeat at Saratoga and their apparent willingness to negotiate peace convinced Vergennes only a permanent alliance could prevent the "disaster" of Anglo-American rapprochement. Assurances of formal French support allowed Congress to reject the Carlisle Peace Commission and insist on nothing short of complete independence.
On February 6, 1778, France and the United States signed the Treaty of Amity and Commerce regulating trade between the two countries, followed by a defensive military alliance against Britain, the Treaty of Alliance. In return for French guarantees of American independence, Congress undertook to defend their interests in the West Indies, while both sides agreed not to make a separate peace; conflict over these provisions would lead to the 1798 to 1800 Quasi-War. Charles III of Spain was invited to join on the same terms but refused, largely due to concerns over the impact of the Revolution on Spanish colonies in the Americas. Spain had complained on multiple occasions about encroachment by American settlers into Louisiana, a problem that could only get worse once the United States replaced Britain. | [
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771_15 | Although Spain ultimately made important contributions to American success, in the Treaty of Aranjuez (1779), Charles agreed only to support France's war with Britain outside America, in return for help in recovering Gibraltar, Menorca and Spanish Florida. The terms were confidential since several conflicted with American aims; for example, the French claimed exclusive control of the Newfoundland cod fisheries, a non-negotiable for colonies like Massachusetts. One less well-known impact of this agreement was the abiding American distrust of 'foreign entanglements'; the US would not sign another treaty until the NATO agreement in 1949. This was because the US had agreed not to make peace without France, while Aranjuez committed France to keep fighting until Spain recovered Gibraltar, effectively making it a condition of US independence without the knowledge of Congress.
To encourage French participation in the struggle for independence, the US representative in Paris, Silas Deane promised promotion and command positions to any French officer who joined the Continental Army. Although many proved incompetent, one outstanding exception was Gilbert du Motier, Marquis de Lafayette, whom Congress appointed a major General. In addition to his military ability, Lafayette showed considerable political skill in building support for Washington among his officers and within Congress, liaising with French army and naval commanders, and promoting the Patriot cause in France.
When the war started, Britain tried to borrow the Dutch-based Scots Brigade for service in America, but pro-Patriot sentiment led the States General to refuse. Although the Republic was no longer a major power, prior to 1774 they still dominated the European carrying trade, and Dutch merchants made large profits shipping French-supplied munitions to the Patriots. This ended when Britain declared war in December 1780, a conflict that proved disastrous to the Dutch economy. The Dutch were also excluded from the First League of Armed Neutrality, formed by Russia, Sweden and Denmark in March 1780 to protect neutral shipping from being stopped and searched for contraband by Britain and France. | [
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771_16 | The British government failed to take into account the strength of the American merchant marine and support from European countries, which allowed the colonies to import munitions and continue trading with relative impunity. While well aware of this, the North administration delayed placing the Royal Navy on a war footing for cost reasons; this prevented the institution of an effective blockade and restricted them to ineffectual diplomatic protests. Traditional British policy was to employ European land-based allies to divert the opposition, a role filled by Prussia in the Seven Years' War; in 1778, they were diplomatically isolated and faced war on multiple fronts.
Meanwhile, George III had given up on subduing America while Britain had a European war to fight. He did not welcome war with France, but he believed the British victories over France in the Seven Years' War as a reason to believe in ultimate victory over France. Britain could not find a powerful ally among the Great Powers to engage France on the European continent. Britain subsequently changed its focus into the Caribbean theater, and diverted major military resources away from America.
Vergennes colleague "For her honour, France had to seize this opportunity to rise from her degradation......
"If she neglected it, if fear overcame duty, she would add debasement to humiliation, and become an object of contempt to her own century and to all future peoples".
Stalemate in the North
At the end of 1777, Howe resigned and was replaced by Sir Henry Clinton on May 24, 1778; with French entry into the war, he was ordered to consolidate his forces in New York. On June 18, the British departed Philadelphia with the reinvigorated Americans in pursuit; the Battle of Monmouth on June 28 was inconclusive but boosted Patriot morale. Washington had rallied Charles Lee's broken regiments, the Continentals repulsed British bayonet charges, the British rear guard lost perhaps 50 per-cent more casualties, and the Americans held the field at the end of the day. That midnight, the newly installed Clinton continued his retreat to New York. | [
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771_17 | A French naval force under Admiral Charles Henri Hector d'Estaing was sent to assist Washington; deciding New York was too formidable a target, in August they launched a combined attack on Newport, with General John Sullivan commanding land forces. The resulting Battle of Rhode Island was indecisive; badly damaged by a storm, the French withdrew to avoid putting their ships at risk. Further activity was limited to British raids on Chestnut Neck and Little Egg Harbor in October.
In July 1779, the Americans captured British positions at Stony Point and Paulus Hook. Clinton unsuccessfully tried to tempt Washington into a decisive engagement by sending General William Tryon to raid Connecticut. In July, a large American naval operation, the Penobscot Expedition, attempted to retake Maine, then part of Massachusetts, but was defeated. Persistent Iroquois raids along the border with Quebec led to the punitive Sullivan Expedition in April 1779, destroying many settlements but failing to stop them.
During the winter of 1779–1780, the Continental Army suffered greater hardships than at Valley Forge. Morale was poor, public support fell away in the long war, the Continental dollar was virtually worthless, the army was plagued with supply problems, desertion was common, and mutinies occurred in the Pennsylvania Line and New Jersey Line regiments over the conditions in early 1780.
In June 1780, Clinton sent 6,000 men under Wilhelm von Knyphausen to retake New Jersey, but they were halted by local militia at the Battle of Connecticut Farms; although the Americans withdrew, Knyphausen felt he was not strong enough to engage Washington's main force and retreated. A second attempt two weeks later ended in a British defeat at the Battle of Springfield, effectively ending their ambitions in New Jersey. In July, Washington appointed Benedict Arnold commander of West Point; his attempt to betray the fort to the British failed due to incompetent planning, and the plot was revealed when his British contact John André was captured and later executed. Arnold escaped to New York and switched sides, an action justified in a pamphlet addressed "To the Inhabitants of America"; the Patriots condemned his betrayal, while he found himself almost as unpopular with the British. | [
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771_18 | The war to the west of the Appalachians was largely confined to skirmishing and raids. In February 1778, an expedition of militia to destroy British military supplies in settlements along the Cuyahoga River was halted by adverse weather. Later in the year, a second campaign was undertaken to seize the Illinois Country from the British. Virginia militia, Canadien settlers, and Indian allies commanded by Colonel George Rogers Clark captured Kaskaskia on July 4 then secured Vincennes, though Vincennes was recaptured by Quebec Governor Henry Hamilton. In early 1779, the Virginians counterattacked in the siege of Fort Vincennes and took Hamilton prisoner. Clark secured western British Quebec as the American Northwest Territory in the Treaty of Paris concluding the war.
On May 25, 1780, British Colonel Henry Bird invaded Kentucky as part of a wider operation to clear American resistance from Quebec to the Gulf coast. Their Pensacola advance on New Orleans was overcome by Spanish Governor Gálvez's offensive on Mobile. Simultaneous British attacks were repulsed on St. Louis by the Spanish Lieutenant Governor de Leyba, and on the Virginia county courthouse at Cahokia by Lieutenant Colonel Clark. The British initiative under Bird from Detroit was ended at the rumored approach of Clark. The scale of violence in the Licking River Valley, such as during the Battle of Blue Licks, was extreme "even for frontier standards". It led to men of English and German settlements to join Clark's militia when the British and their auxiliaries withdrew to the Great Lakes. The Americans responded with a major offensive along the Mad River in August which met with some success in the Battle of Piqua but did not end Indian raids.
French soldier Augustin de La Balme led a Canadian militia in an attempt to capture Detroit, but they dispersed when Miami natives led by Little Turtle attacked the encamped settlers on November 5. The war in the west had become a stalemate with the British garrison sitting in Detroit and the Virginians expanding westward settlements north of the Ohio River in the face of British-allied Indian resistance.
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771_19 | War in the South
The "Southern Strategy" was developed by Lord Germain, based on input from London-based Loyalists like Joseph Galloway. They argued it made no sense to fight the Patriots in the north where they were strongest, while the New England economy was reliant on trade with Britain, regardless of who governed it. On the other hand, duties on tobacco made the South far more profitable for Britain, while local support meant securing it required small numbers of regular troops. Victory would leave a truncated United States facing British possessions in the south, Canada to the north, and Ohio on their western border; with the Atlantic seaboard controlled by the Royal Navy, Congress would be forced to agree to terms. However, assumptions about the level of Loyalist support proved wildly optimistic.
Germain accordingly ordered Augustine Prévost, the British commander in East Florida, to advance into Georgia in December 1778. Lieutenant-Colonel Archibald Campbell, an experienced officer taken prisoner earlier in the war before being exchanged for Ethan Allen, captured Savannah on December 29, 1778. He recruited a Loyalist militia of nearly 1,100, many of whom allegedly joined only after Campbell threatened to confiscate their property. Poor motivation and training made them unreliable troops, as demonstrated in their defeat by Patriot militia at the Battle of Kettle Creek on February 14, 1779, although this was offset by British victory at Brier Creek on March 3.
In June, Prévost launched an abortive assault on Charleston, before retreating to Savannah, an operation notorious for widespread looting by British troops that enraged both Loyalists and Patriots. In October, a joint French and American operation under Admiral d'Estaing and General Benjamin Lincoln failed to recapture Savannah. Prévost was replaced by Lord Cornwallis, who assumed responsibility for Germain's strategy; he soon realized estimates of Loyalist support were considerably over-stated, and he needed far larger numbers of regular forces.
Reinforced by Clinton, his troops captured Charleston in May 1780, inflicting the most serious Patriot defeat of the war; over 5,000 prisoners were taken and the Continental Army in the south effectively destroyed. On May 29, Loyalist regular Banastre Tarleton defeated an American force of 400 at the Battle of Waxhaws; over 120 were killed, many allegedly after surrendering. Responsibility is disputed, Loyalists claiming Tarleton was shot at while negotiating terms of surrender, but it was later used as a recruiting tool by the Patriots. | [
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771_20 | Clinton returned to New York, leaving Cornwallis to oversee the south; despite their success, the two men left barely on speaking terms, with dire consequences for the future conduct of the war. The Southern strategy depended on local support, but this was undermined by a series of coercive measures. Previously, captured Patriots were sent home after swearing not to take up arms against the king; they were now required to fight their former comrades, while the confiscation of Patriot-owned plantations led formerly neutral "grandees" to side with them. Skirmishes at Williamson's Plantation, Cedar Springs, Rocky Mount, and Hanging Rock signaled widespread resistance to the new oaths throughout South Carolina.
In July, Congress appointed General Horatio Gates commander in the south; he was defeated at the Battle of Camden on August 16, leaving Cornwallis free to enter North Carolina. Despite battlefield success, the British could not control the countryside and Patriot attacks continued; before moving north, Cornwallis sent Loyalist militia under Major Patrick Ferguson to cover his left flank, leaving their forces too far apart to provide mutual support. In early October, Ferguson was defeated at the Battle of Kings Mountain, dispersing organized Loyalist resistance in the region. Despite this, Cornwallis continued into North Carolina hoping for Loyalist support, while Washington replaced Gates with General Nathanael Greene in December 1780.
Greene divided his army, leading his main force southeast pursued by Cornwallis; a detachment was sent southwest under Daniel Morgan, who defeated Tarleton's British Legion at Cowpens on January 17, 1781, nearly eliminating it as a fighting force. The Patriots now held the initiative in the south, with the exception of a raid on Richmond led by Benedict Arnold in January 1781. Greene led Cornwallis on a series of countermarches around North Carolina; by early March, the British were exhausted and short of supplies and Greene felt strong enough to fight the Battle of Guilford Court House on March 15. Although victorious, Cornwallis suffered heavy casualties and retreated to Wilmington, North Carolina seeking supplies and reinforcements. | [
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771_21 | The Patriots now controlled most of the Carolinas and Georgia outside the coastal areas; after a minor reversal at the Battle of Hobkirk's Hill, they recaptured Fort Watson and Fort Motte on April 15. On June 6, Brigadier General Andrew Pickens captured Augusta, leaving the British in Georgia confined to Charleston and Savannah. The assumption Loyalists would do most of the fighting left the British short of troops and battlefield victories came at the cost of losses they could not replace. Despite halting Greene's advance at the Battle of Eutaw Springs on September 8, Cornwallis withdrew to Charleston with little to show for his campaign.
Western campaign
When Spain joined France's war against Britain in 1779, their treaty specifically excluded Spanish military action in North America. However, from the beginning of the war, Bernardo de Gálvez, the Governor of Spanish Louisiana, allowed the Americans to import supplies and munitions into New Orleans, then ship them to Pittsburgh. This provided an alternative transportation route for the Continental Army, bypassing the British blockade of the Atlantic Coast.
The trade was organized by Oliver Pollock, a successful merchant in Havana and New Orleans who was appointed US "commercial agent". It also helped support the American campaign in the west; in the 1778 Illinois campaign, militia under General George Rogers Clark cleared the British from what was then part of Quebec, creating Illinois County, Virginia.
Despite official neutrality, Gálvez initiated offensive operations against British outposts. First, he cleared British garrisons in Baton Rouge, Louisiana, Fort Bute, and Natchez, Mississippi, and captured five forts. In doing so, Gálvez opened navigation on the Mississippi River north to the American settlement in Pittsburg.
In 1781, Galvez and Pollock campaigned east along the Gulf Coast to secure West Florida, including British-held Mobile and Pensacola. The Spanish operations crippled the British supply of armaments to British Indian allies, which effectively suspended a military alliance to attack settlers between the Mississippi River and the Appalachian Mountains.
British defeat in the United States | [
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771_22 | British defeat in the United States
Clinton spent most of 1781 based in New York City; he failed to construct a coherent operational strategy, partly due to his difficult relationship with Admiral Marriot Arbuthnot. In Charleston, Cornwallis independently developed an aggressive plan for a campaign in Virginia, which he hoped would isolate Greene's army in the Carolinas and cause the collapse of Patriot resistance in the South. This was approved by Lord Germain in London, but neither of them informed Clinton.
Washington and Rochambeau now discussed their options; the former wanted to attack New York, the latter Virginia, where Cornwallis' forces were less well-established and thus easier to defeat. Washington eventually gave way and Lafayette took a combined Franco-American force into Virginia, but Clinton misinterpreted his movements as preparations for an attack on New York. Concerned by this threat, he instructed Cornwallis to establish a fortified sea base where the Royal Navy could evacuate his troops to help defend New York.
When Lafayette entered Virginia, Cornwallis complied with Clinton's orders and withdrew to Yorktown, where he constructed strong defenses and awaited evacuation. An agreement by the Spanish navy to defend the French West Indies allowed Admiral de Grasse to relocate to the Atlantic seaboard, a move Arbuthnot did not anticipate. This provided Lafayette naval support, while the failure of previous combined operations at Newport and Savannah meant their co-ordination was planned more carefully. Despite repeated urging from his subordinates, Cornwallis made no attempt to engage Lafayette before he could establish siege lines. Even worse, expecting to be withdrawn within a few days he abandoned the outer defenses, which were promptly occupied by the besiegers and hastened British defeat. | [
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771_23 | On August 31, a British fleet under Thomas Graves left New York for Yorktown. After landing troops and munitions for the besiegers on August 30, de Grasse had remained in Chesapeake Bay and intercepted him on September 5; although the Battle of the Chesapeake was indecisive in terms of losses, Graves was forced to retreat, leaving Cornwallis isolated. An attempted breakout over the York River at Gloucester Point failed due to bad weather. Under heavy bombardment with dwindling supplies, Cornwallis felt his situation was hopeless and on October 16 sent emissaries to Washington to negotiate surrender; after twelve hours of negotiations, these were finalized the next day. Responsibility for defeat was the subject of fierce public debate between Cornwallis, Clinton and Germain. Despite criticism from his junior officers, Cornwallis retained the confidence of his peers and later held a series of senior government positions; Clinton ultimately took most of the blame and spent the rest of his life in obscurity.
Subsequent to Yorktown, American forces were assigned to supervise the armistice between Washington and Clinton made to facilitate British departure following the January 1782 law of Parliament forbidding any further British offensive action in North America. British-American negotiations in Paris led to preliminaries signed November 1782 acknowledging US independence. The enacted Congressional war aim for British withdrawal from its North American claims to be ceded to the US was completed for the coastal cities in stages.
In the South, Generals Greene and Wayne loosely invested the withdrawing British at Savanna and Charleston. There they observed the British finally taking off their regulars from Charleston December 14, 1782. Loyalist provincial militias of whites and free blacks, as well as Loyalists with their slaves were transported in a relocation to Nova Scotia and the British Caribbean. Native American allies of the British and some freed blacks were left to escape through the American lines unaided.
Washington moved his army to New Windsor on the Hudson River about sixty miles north of New York City, and there the substance of the American army was furloughed home with officers at half pay until the Treaty of Paris formally ended the war on September 3, 1783. At that time, Congress decommissioned the regiments of Washington's Continental Army and began issuing land grants to veterans in the Northwest Territories for their war service. The last of the British occupation of New York City ended on November 25, 1783, with the departure of Clinton's replacement, General Sir Guy Carleton.
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771_24 | Strategy and commanders
To win their insurrection, the Americans needed to outlast the British will to continue the fight. To restore the empire, the British had to defeat the Continental Army in the early months, and compel the Congress to dissolve itself. Historian Terry M. Mays identifies three separate types of warfare, the first being a colonial conflict in which objections to Imperial trade regulation were as significant as taxation policy. The second was a civil war with all thirteen states split between Patriots, Loyalists and those who preferred to remain neutral. Particularly in the south, many battles were fought between Patriots and Loyalists with no British involvement, leading to divisions that continued after independence was achieved.
The third element was a global war between France, Spain, the Dutch Republic and Britain, with America as one of a number of different theaters. After entering the war in 1778, France provided the Americans money, weapons, soldiers, and naval assistance, while French troops fought under US command in North America. While Spain did not formally join the war in America, they provided access to the Mississippi River and by capturing British possessions on the Gulf of Mexico denied bases to the Royal Navy, as well as retaking Menorca and besieging Gibraltar in Europe.
Although the Dutch Republic was no longer a major power, prior to 1774 they still dominated the European carrying trade, and Dutch merchants made large profits by shipping French-supplied munitions to the Patriots. This ended when Britain declared war in December 1780 and the conflict proved disastrous to their economy. The Dutch were also excluded from the First League of Armed Neutrality, formed by Russia, Sweden and Denmark in March 1780 to protect neutral shipping from being stopped and searched for contraband by Britain and France. While of limited effect, these interventions forced the British to divert men and resources away from North America.
American strategy
Congress had multiple advantages if the rebellion turned into a protracted war. Their prosperous state populations depended on local production for food and supplies rather than on imports from their mother country that lay six to twelve weeks away by sail. They were spread across most of the North American Atlantic seaboard, stretching 1,000 miles. Most farms were remote from the seaports, and controlling four or five major ports did not give British armies control over the inland areas. Each state had established internal distribution systems. | [
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771_25 | Each former colony had a long-established system of local militia, combat-tested in support of British regulars thirteen years before to secure an expanded British Empire. Together they took away French claims in North America west to the Mississippi River in the French and Indian War. The state legislatures independently funded and controlled their local militias. In the American Revolution, they trained and provided Continental Line regiments to the regular army, each with their own state officer corps. Motivation was also a major asset: each colonial capital had its own newspapers and printers, and the Patriots had more popular support than the Loyalists. British hoped that the Loyalists would do much of the fighting, but they fought less than expected.
Continental Army
When the war began, Congress lacked a professional army or navy, and each colony only maintained local militias. Militiamen were lightly armed, had little training, and usually did not have uniforms. Their units served for only a few weeks or months at a time and lacked the training and discipline of more experienced soldiers. Local county militias were reluctant to travel far from home and they were unavailable for extended operations. To compensate for this, Congress established a regular force known as the Continental Army on June 14, 1775, the origin of the modern United States Army, and appointed Washington as commander-in-chief. However, it suffered significantly from the lack of an effective training program and from largely inexperienced officers and sergeants, offset by a few senior officers.
Each state legislature appointed officers for both county and state militias and their regimental Continental Line officers; although Washington was required to accept Congressional appointments, he was still permitted to choose and command his own generals, such as Nathanael Greene, his chief of artillery, Henry Knox, and Alexander Hamilton, the chief of staff. One of Washington's most successful recruits to general officer was Baron Friedrich Wilhelm von Steuben, a veteran of the Prussian general staff who wrote the Revolutionary War Drill Manual. The development of the Continental Army was always a work in progress and Washington used both his regulars and state militia throughout the war; when properly employed, the combination allowed them to overwhelm smaller British forces, as at Concord, Boston, Bennington, and Saratoga. Both sides used partisan warfare, but the state militias effectively suppressed Loyalist activity when British regulars were not in the area. | [
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771_26 | Washington designed the overall military strategy of the war in cooperation with Congress, established the principle of civilian supremacy in military affairs, personally recruited his senior officer corps, and kept the states focused on a common goal. For the first three years until after Valley Forge, the Continental Army was largely supplemented by local state militias. Initially, Washington employed the inexperienced officers and untrained troops in Fabian strategies rather than risk frontal assaults against Britain's professional soldiers and officers. Over the course of the entire war, Washington lost more battles than he won, but he never surrendered his troops and maintained a fighting force in the face of British field armies and never gave up fighting for the American cause.
By prevailing European standards, the armies in America were relatively small, limited by lack of supplies and logistics; the British in particular were constrained by the difficulty of transporting troops across the Atlantic and dependence on local supplies. Washington never directly commanded more than 17,000 men, while the combined Franco-American army at Yorktown was only about 19,000. At the beginning of 1776, Patriot forces consisted of 20,000 men, with two-thirds in the Continental Army and the other third in the various state militias. About 250,000 men served as regulars or as militia for the Revolutionary cause over eight years during wartime, but there were never more than 90,000 men under arms at one time.
As a whole, American officers never equaled their opponents in tactics and maneuvers, and they lost most of the pitched battles. The great successes at Boston (1776), Saratoga (1777), and Yorktown (1781) were won from trapping the British far from base with a greater number of troops. Nevertheless, after 1778, Washington's army was transformed into a more disciplined and effective force, mostly by Baron von Steuben's training. Immediately after the Army emerged from Valley Forge, it proved its ability to match the British troops in action at the Battle of Monmouth, including a black Rhode Island regiment fending off a British bayonet attack then counter-charging for the first time in Washington's army. Here Washington came to realize that saving entire towns was not necessary, but preserving his army and keeping the revolutionary spirit alive was more important in the long run. Washington informed Henry Laurens "that the possession of our towns, while we have an army in the field, will avail them little." | [
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771_27 | Although Congress was responsible for the war effort and provided supplies to the troops, Washington took it upon himself to pressure the Congress and state legislatures to provide the essentials of war; there was never nearly enough. Congress evolved in its committee oversight and established the Board of War, which included members of the military. Because the Board of War was also a committee ensnared with its own internal procedures, Congress also created the post of Secretary of War, and appointed Major General Benjamin Lincoln in February 1781 to the position. Washington worked closely with Lincoln to coordinate civilian and military authorities and took charge of training and supplying the army.
Continental Navy
During the first summer of the war, Washington began outfitting schooners and other small seagoing vessels to prey on ships supplying the British in Boston. Congress established the Continental Navy on October 13, 1775, and appointed Esek Hopkins as its first commander; for most of the war, it consisted of a handful of small frigates and sloops, supported by numerous privateers. On November 10, 1775, Congress authorized the creation of the Continental Marines, forefather of the United States Marine Corps.
John Paul Jones became the first American naval hero by capturing HMS Drake on April 24, 1778, the first victory for any American military vessel in British waters. The last was by the frigate USS Alliance commanded by Captain John Barry. On March 10, 1783, the Alliance outgunned HMS Sybil in a 45-minute duel while escorting Spanish gold from Havana to Congress. After Yorktown, all US Navy ships were sold or given away; it was the first time in America's history that it had no fighting forces on the high seas.
Congress primarily commissioned privateers to reduce costs and to take advantage of the large proportion of colonial sailors found in the British Empire. Overall, they included 1,700 ships that successfully captured 2,283 enemy ships to damage the British effort and to enrich themselves with the proceeds from the sale of cargo and the ship itself. About 55,000 sailors served aboard American privateers during the war.
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771_28 | France
At the beginning of the war, the Americans had no major international allies, as most nation-states watched and waited to see how developments would unfold in British North America. Over time, the Continental Army acquitted itself well in the face of British regulars and their German auxiliaries known to all European great powers. Battles such as the Battle of Bennington, the Battles of Saratoga, and even defeats such as the Battle of Germantown, proved decisive in gaining the attention and support of powerful European nations including France and Spain, and the Dutch Republic; the latter moved from covertly supplying the Americans with weapons and supplies to overtly supporting them.
The decisive American victory at Saratoga convinced France, who was already a long-time rival of Britain, to offer the Americans the Treaty of Amity and Commerce. The two nations also agreed to a defensive Treaty of Alliance to protect their trade and also guaranteed American independence from Britain. To engage the United States as a French ally militarily, the treaty was conditioned on Britain initiating a war on France to stop it from trading with the US. Spain and the Dutch Republic were invited to join by both France and the United States in the treaty, but neither made a formal reply.
On June 13, 1778, France declared war on Great Britain, and it invoked the French military alliance with the US, which ensured additional US privateer support for French possessions in the Caribbean. Washington worked closely with the soldiers and navy that France would send to America, primarily through Lafayette on his staff. French assistance made critical contributions required to defeat General Charles Cornwallis at Yorktown in 1781.
British strategy
The British military had considerable experience of fighting in North America, most recently during the Seven Years' War which forced France to give up New France in 1763. However, in previous conflicts they benefited from local logistics, as well as support from the colonial militia, which was not available in the American Revolutionary War. Reinforcements had to come from Europe, and maintaining large armies over such distances was extremely complex; ships could take three months to cross the Atlantic, and orders from London were often outdated by the time they arrived. | [
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771_29 | Prior to the conflict, the colonies were largely autonomous economic and political entities, with no centralized area of ultimate strategic importance. This meant that, unlike Europe where the fall of a capital city often ended wars, that in America continued even after the loss of major settlements such as Philadelphia, the seat of Congress, New York and Charleston. British power was reliant on the Royal Navy, whose dominance allowed them to resupply their own expeditionary forces while preventing access to enemy ports. However, the majority of the American population was agrarian, rather than urban; supported by the French navy and blockade runners based in the Dutch Caribbean, their economy was able to survive.
The geographical size of the colonies and limited manpower meant the British could not simultaneously conduct military operations and occupy territory without local support. Debate persists over whether their defeat was inevitable; one British statesman described it as "like trying to conquer a map". While Ferling argues Patriot victory was nothing short of a miracle, Ellis suggests the odds always favored the Americans, especially after Howe squandered the chance of a decisive British success in 1776, an "opportunity that would never come again". The US military history speculates the additional commitment of 10,000 fresh troops in 1780 would have placed British victory "within the realm of possibility".
British Army
The expulsion of France from North America in 1763 led to a drastic reduction in British troop levels in the colonies; in 1775, there were only 8,500 regular soldiers among a civilian population of 2.8 million. The bulk of military resources in the Americas were focused on defending sugar islands in the Caribbean; Jamaica alone generated more revenue than all thirteen American colonies combined. With the end of the Seven Years' War, the permanent army in Britain was also cut back, which resulted in administrative difficulties when the war began a decade later. | [
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771_30 | Over the course of the war, there were four separate British commanders-in-chief, the first of whom was Thomas Gage; appointed in 1763, his initial focus was establishing British rule in former French areas of Canada. Rightly or wrongly, many in London blamed the revolt on his failure to take firm action earlier, and he was relieved after the heavy losses incurred at Bunker Hill. His replacement was Sir William Howe, a member of the Whig faction in Parliament who opposed the policy of coercion advocated by Lord North; Cornwallis, who later surrendered at Yorktown, was one of many senior officers who initially refused to serve in North America.
The 1775 campaign showed the British overestimated the capabilities of their own troops and underestimated the colonial militia, requiring a reassessment of tactics and strategy. However, it allowed the Patriots to take the initiative and British authorities rapidly lost control over every colony. Howe's responsibility is still debated; despite receiving large numbers of reinforcements, Bunker Hill seems to have permanently affected his self-confidence and lack of tactical flexibility meant he often failed to follow up opportunities. Many of his decisions were attributed to supply problems, such as the delay in launching the New York campaign and failure to pursue Washington's beaten army. Having lost the confidence of his subordinates, he was recalled after Burgoyne surrendered at Saratoga.
Following the failure of the Carlisle Commission, British policy changed from treating the Patriots as subjects who needed to be reconciled to enemies who had to be defeated. In 1778, Howe was replaced by Sir Henry Clinton, appointed instead of Carleton who was considered overly cautious. Regarded as an expert on tactics and strategy, like his predecessors Clinton was handicapped by chronic supply issues. As a result, he was largely inactive in 1779 and much of 1780; in October 1780, he warned Germain of "fatal consequences" if matters did not improve.
In addition, Clinton's strategy was compromised by conflict with political superiors in London and his colleagues in North America, especially Admiral Mariot Arbuthnot, replaced in early 1781 by Rodney. He was neither notified nor consulted when Germain approved Cornwallis' invasion of the south in 1781 and delayed sending him reinforcements believing the bulk of Washington's army was still outside New York City. After the surrender at Yorktown, Clinton was relieved by Carleton, whose major task was to oversee the evacuation of Loyalists and British troops from Savannah, Charleston, and New York City.
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771_31 | German Troops
During the 18th century, all states commonly hired foreign soldiers, especially Britain; during the Seven Years' War, they comprised 10% of the British army and their use caused little debate. When it became clear additional troops were needed to suppress the revolt in America, it was decided to employ mercenaries. There were several reasons for this, including public sympathy for the Patriot cause, an historical reluctance to expand the British army and the time needed to recruit and train new regiments. An alternate source was readily available in the Holy Roman Empire, where many smaller states had a long tradition of renting their armies to the highest bidder. The most important was Hesse-Cassel, known as "the Mercenary State".
The first supply agreements were signed by the North administration in late 1775; over the next decade, more than 40,000 Germans fought in North America, Gibraltar, South Africa and India, of whom 30,000 served in the American War. Often generically referred to as "Hessians", they included men from many other states, including Hanover and Brunswick. Sir Henry Clinton recommended recruiting Russian troops whom he rated very highly, having seen them in action against the Ottomans; however, negotiations with Catherine the Great made little progress.
Unlike previous wars their use led to intense political debate in Britain, France, and even Germany, where Frederick the Great refused to provide passage through his territories for troops hired for the American war. In March 1776, the agreements were challenged in Parliament by Whigs who objected to "coercion" in general, and the use of foreign soldiers to subdue "British subjects". The debates were covered in detail by American newspapers, which reprinted key speeches and in May 1776 they received copies of the treaties themselves. Provided by British sympathizers, these were smuggled into North America from London by George Merchant, a recently released American prisoner.
The prospect of mercenaries being used in the colonies bolstered support for independence, more so than taxation and other acts combined; the King was accused of declaring war on his own subjects, leading to the idea there were now two separate governments. By apparently showing Britain was determined to go to war, it made hopes of reconciliation seem naive and hopeless, while the employment of 'foreign mercenaries' became one of the charges levelled against George III in the Declaration of Independence. The Hessian reputation within Germany for brutality also increased support for the Patriot cause among German-American immigrants. | [
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771_32 | The presence of over 150,000 German-Americans meant both sides felt these mercenaries might be persuaded to desert; one reason Clinton suggested employing Russians was that he felt they were less likely to defect. When the first German troops arrived on Staten Island in August 1776, Congress approved the printing of "handbills" promising land and citizenship to any willing to join the Patriot cause. The British launched a counter-campaign claiming deserters could well be executed for meddling in a war that was not theirs. Desertion among the Germans occurred throughout the war, with the highest rate of desertion occurring during the time between the surrender at Yorktown and the Treaty of Paris. German regiments were central to the British war effort; of the estimated 30,000 sent to America, some 13,000 became casualties.
Revolution as civil war
Loyalists
Wealthy Loyalists convinced the British government that most of the colonists were sympathetic toward the Crown; consequently, British military planners relied on recruiting Loyalists, but had trouble recruiting sufficient numbers as the Patriots had widespread support. Nevertheless, they continued to deceive themselves on their level of American support as late as 1780, a year before hostilities ended.
Approximately 25,000 Loyalists fought for the British throughout the war. Although Loyalists constituted about twenty percent of the colonial population, they were concentrated in distinct communities. Many of them lived among large plantation owners in the Tidewater region and South Carolina who produced cash crops in tobacco and indigo comparable to global markets in Caribbean sugar.
When the British began probing the backcountry in 1777–1778, they were faced with a major problem: any significant level of organized Loyalist activity required a continued presence of British regulars. The available manpower that the British had in America was insufficient to protect Loyalist territory and counter American offensives. The Loyalist militias in the South were constantly defeated by neighboring Patriot militia. The most critical combat between the two partisan militias was at the Battle of Kings Mountain; the Patriot victory irreversibly crippled any further Loyalist militia capability in the South. | [
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771_33 | When the early war policy was administered by General William Howe, the Crown's need to maintain Loyalist support prevented it from using the traditional revolt suppression methods. The British cause suffered when their troops ransacked local homes during an aborted attack on Charleston in 1779 that enraged both Patriots and Loyalists. After Congress rejected the Carlisle Peace Commission in 1778 and Westminster turned to "hard war" during Clinton's command, neutral colonists in the Carolinas often allied with the Patriots whenever brutal combat broke out between Tories and Whigs. Conversely, Loyalists gained support when Patriots intimidated suspected Tories by destroying property or tarring and feathering.
A Loyalist militia unit—the British Legion—provided some of the best troops in British service that it received a commission in the British Army: it was a mixed regiment of 250 dragoons and 200 infantry supported by batteries of flying artillery. It was commanded by Banastre Tarleton and gained a fearsome reputation in the colonies for "brutality and needless slaughter". In May 1779 the British Legion was one of five regiments that formed the American Establishment.
Women
Women played various roles during the Revolutionary War; they often accompanied their husbands when permitted to do so. For example, throughout the war Martha Washington was known to visit and provide aid to her husband George at various American camps, and Frederika Charlotte Riedesel documented the Saratoga campaign. Women often accompanied armies as camp followers to sell goods and perform necessary tasks in hospitals and camps. They were a necessary part of eighteenth-century armies, and numbered in the thousands during the war.
Women also assumed military roles: aside from auxiliary tasks like treating the wounded or setting up camp, some dressed as men to directly support combat, fight, or act as spies on both sides of the Revolutionary War. Anna Maria Lane joined her husband in the Army and wore men's clothes by the time the Battle of Germantown happened. The Virginia General Assembly later cited her bravery: she fought while dressed as a man and "performed extraordinary military services, and received a severe wound at the battle of Germantown ... with the courage of a soldier". | [
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771_34 | On April 26, 1777, Sybil Ludington rode to alert militia forces of Putnam County, New York, and Danbury, Connecticut, to warn them of the British's approach; she has been called the "female Paul Revere". A few others disguised themselves as men. Deborah Sampson fought until her gender was discovered and discharged as a result; Sally St. Clair was killed in action during the war.
African Americans
When war began, the population of the Thirteen Colonies included an estimated 500,000 slaves, predominantly used as labor on Southern plantations. In November 1775, Lord Dunmore, the Royal Governor of Virginia, issued a proclamation that promised freedom to any Patriot-owned slaves willing to bear arms. Although the announcement helped to fill a temporary manpower shortage, white Loyalist prejudice meant recruits were eventually redirected to non-combatant roles. The Loyalists' motive was to deprive Patriot planters of labor rather than to end slavery; Loyalist-owned slaves were returned.
The 1779 Philipsburg Proclamation issued by Clinton extended the offer of freedom to Patriot-owned slaves throughout the colonies. It persuaded entire families to escape to British lines, many of which were employed on farms to grow food for the army by removing the requirement for military service. While Clinton organized the Black Pioneers, he also ensured fugitive slaves were returned to Loyalist owners with orders that they were not to be punished for their attempted escape. As the war progressed, service as regular soldiers in British units became increasingly common; black Loyalists formed two regiments of the Charleston garrison in 1783.
Estimates of the numbers who served the British during the war vary from 25,000 to 50,000, excluding those who escaped during wartime. Thomas Jefferson estimated that Virginia may have lost 30,000 slaves in total escapes. In South Carolina, nearly 25,000 slaves (about 30 percent of the enslaved population) either fled, migrated, or died, which significantly disrupted the plantation economies both during and after the war. | [
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771_35 | Black Patriots were barred from the Continental Army until Washington convinced Congress in January 1778 that there was no other way to replace losses from disease and desertion. The 1st Rhode Island Regiment formed in February included former slaves whose owners were compensated; however, only 140 of its 225 soldiers were black and recruitment stopped in June 1788. Ultimately, around 5,000 African-Americans served in the Continental Army and Navy in a variety of roles, while another 4,000 were employed in Patriot militia units, aboard privateers, or as teamsters, servants, and spies. After the war, a small minority received land grants or Congressional pensions in old age; many others were returned to their masters post-war despite earlier promises of freedom.
As a Patriot victory became increasingly likely, the treatment of Black Loyalists became a point of contention; after the surrender of Yorktown in 1781, Washington insisted all escapees be returned but Cornwallis refused. In 1782 and 1783, around 8,000 to 10,000 freed blacks were evacuated by the British from Charleston, Savannah, and New York; some moved onto London, while 3,000 to 4,000 settled in Nova Scotia, where they founded settlements such as Birchtown. White Loyalists transported 15,000 enslaved blacks to Jamaica and the Bahamas. The free Black Loyalists who migrated to the British West Indies included regular soldiers from Dunmore's Ethiopian Regiment, and those from Charleston who helped garrison the Leeward Islands.
Native Americans
Most Native Americans east of the Mississippi River were affected by the war, and many tribes were divided over how to respond to the conflict. A few tribes were friendly with the colonists, but most Natives opposed the union of the Colonies as a potential threat to their territory. Approximately 13,000 Natives fought on the British side, with the largest group coming from the Iroquois tribes who deployed around 1,500 men.
Early in July 1776, Cherokee allies of Britain attacked the short-lived Washington District of North Carolina. Their defeat splintered both Cherokee settlements and people, and was directly responsible for the rise of the Chickamauga Cherokee, who perpetuated the Cherokee–American wars against American settlers for decades after hostilities with Britain ended. | [
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771_36 | Creek and Seminole allies of Britain fought against Americans in Georgia and South Carolina. In 1778, a force of 800 Creeks destroyed American settlements along the Broad River in Georgia. Creek warriors also joined Thomas Brown's raids into South Carolina and assisted Britain during the Siege of Savannah. Many Native Americans were involved in the fight between Britain and Spain on the Gulf Coast and along the British side of the Mississippi River. Thousands of Creeks, Chickasaws, and Choctaws fought in major battles such as the Battle of Fort Charlotte, the Battle of Mobile, and the Siege of Pensacola.
The Iroquois Confederacy was shattered as a result of the American Revolutionary War, whatever side they took; the Seneca, Onondaga, and Cayuga tribes sided with the British; members of the Mohawks fought on both sides; and many Tuscarora and Oneida sided with the Americans. To retaliate against raids on American settlement by Loyalists and their Indian allies, the Continental Army dispatched the Sullivan Expedition on a punitive expedition throughout New York to cripple the Iroquois tribes that had sided with the British. Mohawk leaders Joseph Louis Cook and Joseph Brant sided with the Americans and the British respectively, which further exacerbated the split.
In the western theater of the American Revolutionary War, conflicts between settlers and Native Americans led to lingering distrust. In the 1783 Treaty of Paris, Great Britain ceded control of the disputed lands between the Great Lakes and the Ohio River, but the Indian inhabitants were not a part of the peace negotiations. Tribes in the Northwest Territory joined as the Western Confederacy and allied with the British to resist American settlement, and their conflict continued after the Revolutionary War as the Northwest Indian War.
Britain's "American war" and peace
Changing Prime Ministers
Lord North, Prime Minister since 1770, delegated control of the war in North America to Lord George Germain and the Earl of Sandwich, who was head of the Royal Navy from 1771 to 1782. Defeat at Saratoga in 1777 made it clear the revolt would not be easily suppressed, especially after the Franco-American alliance of February 1778, and French declaration of war in June. With Spain also expected to join the conflict, the Royal Navy needed to prioritize either the war in America or in Europe; Germain advocated the former, Sandwich the latter. | [
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771_37 | British negotiators now proposed a second peace settlement to Congress. The terms presented by the Carlisle Peace Commission included acceptance of the principle of self-government. Parliament would recognize Congress as the governing body, suspend any objectionable legislation, surrender its right to local colonial taxation, and discuss including American representatives in the House of Commons. In return, all property confiscated from Loyalists would be returned, British debts honored, and locally enforced martial law accepted. However, Congress demanded either immediate recognition of independence or the withdrawal of all British troops; they knew the commission were not authorized to accept these, bringing negotiations to a rapid end.
When the commissioners returned to London in November 1778, they recommended a change in policy. Sir Henry Clinton, the new British Commander-in-Chief in America, was ordered to stop treating the rebels as enemies, rather than subjects whose loyalty might be regained. Those standing orders would be in effect for three years until Clinton was relieved.
North initially backed the Southern strategy attempting to exploit divisions between the mercantile north and slave-owning south, but after the defeat of Yorktown, he was forced to accept the fact that this policy had failed. It was clear the war was lost, although the Royal Navy forced the French to relocate their fleet to the Caribbean in November 1781 and resumed a close blockade of American trade. The resulting economic damage and rising inflation meant the US was now eager to end the war, while France was unable to provide further loans; Congress could no longer pay its soldiers.
On February 27, 1782, a Whig motion to end the offensive war in America was carried by 19 votes. North now resigned, obliging the king to invite Lord Rockingham to form a government; a consistent supporter of the Patriot cause, he made a commitment to US independence a condition of doing so. George III reluctantly accepted and the new government took office on March 27, 1782; however, Rockingham died unexpectedly on July 1, and was replaced by Lord Shelburne who acknowledged American independence.
American Congress signs a peace | [
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771_38 | American Congress signs a peace
When Lord Rockingham, the Whig leader and friend of the American cause was elevated to Prime Minister, Congress consolidated its diplomatic consuls in Europe into a peace delegation at Paris. All were experienced in Congressional leadership. The dean of the delegation was Benjamin Franklin of Pennsylvania. He had become a celebrity in the French Court, but he was also an Enlightenment scientist with influence in the courts of European great powers in Prussia, England's former ally, and Austria, a Catholic empire like Spain. Since the 1760s he had been an organizer of British American inter-colony cooperation, and then a colonial lobbyist to Parliament in London. John Adams of Massachusetts had been consul to the Dutch Republic and was a prominent early New England Patriot. John Jay of New York had been consul to Spain and was a past president of the Continental Congress. As consul to the Dutch Republic, Henry Laurens of South Carolina had secured a preliminary agreement for a trade agreement. He had been a successor to John Jay as president of Congress and with Franklin was a member of the American Philosophical Society. Although active in the preliminaries, he was not a signer of the conclusive treaty.
The Whig negotiators for Lord Rockingham and his successor, Prime Minister Lord Shelburne, included long-time friend of Benjamin Franklin from his time in London, David Hartley and Richard Oswald, who had negotiated Laurens' release from the Tower of London. The Preliminary Peace signed on November 30 met four key Congressional demands: independence, territory up to the Mississippi, navigation rights into the Gulf of Mexico, and fishing rights in Newfoundland.
British strategy was to strengthen the US sufficiently to prevent France from regaining a foothold in North America, and they had little interest in these proposals. However, divisions between their opponents allowed them to negotiate separately with each to improve their overall position, starting with the American delegation in September 1782. The French and Spanish sought to improve their position by creating the U.S. dependent on them for support against Britain, thus reversing the losses of 1763. Both parties tried to negotiate a settlement with Britain excluding the Americans; France proposed setting the western boundary of the US along the Appalachians, matching the British 1763 Proclamation Line. The Spanish suggested additional concessions in the vital Mississippi River Basin, but required the cession of Georgia in violation of the Franco-American alliance. | [
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771_39 | Facing difficulties with Spain over claims involving the Mississippi River, and from France who was still reluctant to agree to American independence until all her demands were met, John Jay promptly told the British that he was willing to negotiate directly with them, cutting off France and Spain, and Prime Minister Lord Shelburne, in charge of the British negotiations, agreed. Key agreements for America in obtaining peace included recognition of United States independence, that she would gain all of the area east of the Mississippi River, north of Florida, and south of Canada; the granting of fishing rights in the Grand Banks, off the coast of Newfoundland and in the Gulf of Saint Lawrence; the United States and Great Britain were to each be given perpetual access to the Mississippi River.
An Anglo-American Preliminary Peace was formally entered into in November 1782, and Congress endorsed the settlement on April 15, 1783. It announced the achievement of peace with independence; the "conclusive" treaty was signed on September 2, 1783, in Paris, effective the next day September 3, when Britain signed its treaty with France. John Adams, who helped draft the treaty, claimed it represented "one of the most important political events that ever happened on the globe". Ratified respectively by Congress and Parliament, the final versions were exchanged in Paris the following spring. On 25 November, the last British troops remaining in the US were evacuated from New York to Halifax.
Aftermath
Washington expressed astonishment that the Americans had won a war against a leading world power, referring to the American victory as "little short of a standing miracle". The conflict between British subjects with the Crown against those with the Congress had lasted over eight years from 1775 to 1783. The last uniformed British troops departed their last east coast port cities in Savannah, Charleston, and New York City, by November 25, 1783. That marked the end of British occupation in the new United States.
On April 9, 1783, Washington issued orders that he had long waited to give, that "all acts of hostility" were to cease immediately. That same day, by arrangement with Washington, General Carleton issued a similar order to British troops. British troops, however, were not to evacuate until a prisoner of war exchange occurred, an effort that involved much negotiation and would take some seven months to effect. | [
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771_40 | As directed by a Congressional resolution of May 26, 1783, all non-commissioned officers and enlisted were furloughed "to their homes" until the "definitive treaty of peace", when they would be automatically discharged. The US armies were directly disbanded in the field as of Washington's General Orders on Monday, June 2, 1783. Once the conclusive Treaty of Paris was signed with Britain, Washington resigned as commander-in-chief at Congress, leaving for his Army retirement at Mount Vernon.
Territory
The expanse of territory that was now the United States was ceded from its colonial Mother country alone. It included millions of sparsely settled acres south of the Great Lakes Line between the Appalachian Mountains and the Mississippi River. The tentative colonial migration west became a flood during the years of the Revolutionary War. Virginia's Kentucky County counted 150 men in 1775. By 1790 fifteen years later, it numbered over 73,000 and was seeking statehood in the United States.
Britain's extended post-war policy for the US continued to try to establish an Indian buffer state below the Great Lakes as late as 1814 during the War of 1812. The formally acquired western American lands continued to be populated by a dozen or so American Indian tribes that had been British allies for the most part. Though British forts on their lands had been ceded to either the French or the British prior to the creation of the United States, Natives were not referred to in the British cession to the US.
While tribes were not consulted by the British for the treaty, in practice the British refused to abandon the forts on territory they formally transferred. Instead, they provisioned military allies for continuing frontier raids and sponsored the Northwest Indian War (1785–1795), including erecting an additional British Fort Miami (Ohio). British sponsorship of local warfare on the United States continued until the Anglo-American Jay Treaty went into effect. At the same time, the Spanish also sponsored war within the US by Indian proxies in its Southwest Territory ceded by France to Britain, then Britain to the Americans. | [
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771_41 | Of the European powers with American colonies adjacent to the newly created United States, Spain was most threatened by American independence, and it was correspondingly the most hostile to it. Its territory adjacent to the US was relatively undefended, so Spanish policy developed a combination of initiatives. Spanish soft power diplomatically challenged the British territorial cession west to the Mississippi and the previous northern boundaries of Spanish Florida. It imposed a high tariff on American goods, then blocked American settler access to the port of New Orleans. Spanish hard power extended war alliances and arms to Southwestern Natives to resist American settlement. A former Continental Army General, James Wilkinson settled in Kentucky County Virginia in 1784, and there he fostered settler secession from Virginia during the Spanish-allied Chickamauga Cherokee war. Beginning in 1787, he received pay as Spanish Agent 13, and subsequently expanded his efforts to persuade American settlers west of the Appalachians to secede from the United States, first in the Washington administration, and later again in the Jefferson administration.
Casualties and losses
The total loss of life throughout the conflict is largely unknown. As was typical in wars of the era, diseases such as smallpox claimed more lives than battle. Between 1775 and 1782, a smallpox epidemic broke out throughout North America, killing an estimated 130,000 among all its populations during those years. Historian Joseph Ellis suggests that Washington's decision to have his troops inoculated against the disease was one of his most important decisions.
Up to 70,000 American Patriots died during active military service. Of these, approximately 6,800 were killed in battle, while at least 17,000 died from disease. The majority of the latter died while prisoners of war of the British, mostly in the prison ships in New York Harbor. The number of Patriots seriously wounded or disabled by the war has been estimated from 8,500 to 25,000.
The French suffered 2,112 killed in combat in the United States. The Spanish lost a total of 124 killed and 247 wounded in West Florida.
A British report in 1781 puts their total Army deaths at 6,046 in North America (1775–1779). Approximately 7,774 Germans died in British service in addition to 4,888 deserters; of the former, it is estimated 1,800 were killed in combat.
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] |
771_42 | Legacy
The American Revolution established the United States with its numerous civil liberties and set an example to overthrow both monarchy and colonial governments. The United States has the world's oldest written constitution, and the constitutions of other free countries often bear a striking resemblance to the US Constitution, often word-for-word in places. It inspired the French, Haitian, Latin American Revolutions, and others into the modern era.
Although the Revolution eliminated many forms of inequality, it did little to change the status of women, despite the role they played in winning independence. Most significantly, it failed to end slavery which continued to be a serious social and political issue and caused divisions that would ultimately end in civil war. While many were uneasy over the contradiction of demanding liberty for some, yet denying it to others, the dependence of southern states on slave labor made abolition too great a challenge. Between 1774 and 1780, many of the states banned the importation of slaves, but the institution itself continued.
In 1782, Virginia passed a law permitting manumission and over the next eight years more than 10,000 slaves were given their freedom. With support from Benjamin Franklin, in 1790 the Quakers petitioned Congress to abolish slavery; the number of abolitionist movements greatly increased, and by 1804 all the northern states had outlawed it. However, even many like Adams who viewed slavery as a 'foul contagion' opposed the 1790 petition as a threat to the Union. In 1808, Jefferson passed legislation banning the importation of slaves, but allowed the domestic slave trade to continue, arguing the federal government had no right to regulate individual states. | [
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771_43 | Historiography
A large historiography concerns the reasons the Americans revolted and successfully broke away. The "Patriots", an insulting term used by the British that was proudly adopted by the Americans, stressed the constitutional rights of Englishmen, especially "No taxation without representation." Contemporaries credited the American Enlightenment with laying the intellectual, moral and ethical foundations of the Revolution among the Founding Fathers. Founders referred to the liberalism in the philosophy of John Locke as powerful influences. Although Two Treatises of Government has long been cited as a major influence on American thinkers, historians David Lundberg and Henry F. May demonstrate that Locke's Essay Concerning Human Understanding was far more widely read than were his political Treatises. Historians since the 1960s have emphasized that the Patriot constitutional argument was made possible by the emergence of a sense of American nationalism that united all 13 colonies. In turn, that nationalism was rooted in a Republican value system that demanded consent of the governed and opposed aristocratic control. In Britain itself, republicanism was a fringe view since it challenged the aristocratic control of the British political system. Political power was not controlled by an aristocracy or nobility in the 13 colonies, and instead, the colonial political system was based on the winners of free elections, which were open to the majority of white men. In the analysis of the coming of the Revolution, historians in recent decades have mostly used one of three approaches.
The Atlantic history view places the American story in a broader context, including revolutions in France and Haiti. It tends to reintegrate the historiographies of the American Revolution and the British Empire.
The "new social history" approach looks at community social structure to find cleavages that were magnified into colonial cleavages.
The ideological approach that centers on republicanism in the United States. Republicanism dictated there would be no royalty, aristocracy or national church but allowed for continuation of the British common law, which American lawyers and jurists understood and approved and used in their everyday practice. Historians have examined how the rising American legal profession adopted British common law to incorporate republicanism by selective revision of legal customs and by introducing more choices for courts. | [
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] |
771_44 | Commemorations of the Revolutionary War
After the first U.S. postage stamp was issued in 1849, the U.S. Post Office frequently issued commemorative stamps celebrating the various people and events of the Revolutionary War. However, it would be more than 140 years after the Revolution before any stamp commemorating that war itself was ever issued. The first such stamp was the 'Liberty Bell' issue of 1926.
See also
1776 in the United States: events, births, deaths, and other years
Timeline of the American Revolution
Topics of the Revolution
Committee of safety (American Revolution)
Financial costs of the American Revolutionary War
Flags of the American Revolution
Naval operations in the American Revolutionary War
Social history of the Revolution
Black Patriot
Christianity in the United States#American Revolution
The Colored Patriots of the American Revolution
History of Poles in the United States#American Revolution
List of clergy in the American Revolution
List of Patriots (American Revolution)
Quakers in the American Revolution
Scotch-Irish Americans#American Revolution
Others in the American Revolution
Nova Scotia in the American Revolution
Watauga Association
Lists of Revolutionary military
List of American Revolutionary War battles
List of British Forces in the American Revolutionary War
List of Continental Forces in the American Revolutionary War
List of infantry weapons in the American Revolution
List of United States militia units in the American Revolutionary War
"Thirteen Colony" economy
Economic history of the US: Colonial economy to 1780
Shipbuilding in the American colonies
Slavery in the United States
Legacy and related
American Revolution Statuary
Commemoration of the American Revolution
Founders Online
Independence Day (United States)
The Last Men of the Revolution
List of plays and films about the American Revolution
Museum of the American Revolution
Tomb of the Unknown Soldier of the American Revolution
United States Bicentennial
List of wars of independence
Bibliographies
Bibliography of the American Revolutionary War
Bibliography of Thomas Jefferson
Bibliography of George Washington
Notes
Citations
Year dates enclosed in [brackets] denote year of original printing
Sources
Britannica.com
Dictionary of American Biography
Encyclopædia Britannica
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] |
771_45 | , p. 73
– Highly regarded examination of British strategy and leadership. An introduction by John W. Shy with his biographical sketch of Mackesy.
Robinson Library
(See also:British Warships in the Age of Sail)
Websites without authors
Canada's Digital Collections Program
History.org
Maryland State House
The History Place
Totallyhistory.com
U.S. Merchant Marine
U.S. National Archives
Valley Forge National Historic Park
Yale Law School, Massachusetts Act
Bibliography
A selection of works relating to the war not listed above; | [
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] |
771_46 | A selection of works relating to the war not listed above;
Allison, David, and Larrie D. Ferreiro, eds. The American Revolution: A World War (Smithsonian, 2018) excerpt
Volumes committed to the American Revolution: Vol. 7; Vol. 8; Vol. 9; Vol. 10
Bobrick, Benson. Angel in the Whirlwind: The Triumph of the American Revolution. Penguin, 1998 (paperback reprint)
Chartrand, Rene. The French Army in the American War of Independence (1994). Short (48pp), very well illustrated descriptions.
Commager, Henry Steele and Richard B. Morris, eds. The Spirit of 'Seventy-Six': The Story of the American Revolution as told by Participants. (Indianapolis: Bobbs-Merrill, 1958). online
Conway, Stephen. The War of American Independence 1775–1783. Publisher: E. Arnold, 1995. . 280 pp.
Kwasny, Mark V. Washington's Partisan War, 1775–1783. Kent, Ohio: 1996. . Militia warfare.
Library of Congress
May, Robin. The British Army in North America 1775–1783 (1993). Short (48pp), very well illustrated descriptions.
National Institute of Health
Neimeyer, Charles Patrick. America Goes to War: A Social History of the Continental Army (1995)
Royal Navy Museum
Stoker, Donald, Kenneth J. Hagan, and Michael T. McMaster, eds. Strategy in the American War of Independence: a global approach (Routledge, 2009) excerpt.
Symonds, Craig L. A Battlefield Atlas of the American Revolution (1989), newly drawn maps emphasizing the movement of military units
U.S. Army, "The Winning of Independence, 1777–1783" American Military History Volume I, 2005.
U.S. National Park Service
Zlatich, Marko; Copeland, Peter. General Washington's Army (1): 1775–78 (1994). Short (48pp), very well illustrated descriptions.
——. General Washington's Army (2): 1779–83 (1994). Short (48pp), very well illustrated descriptions. | [
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771_47 | Primary sources
In addition to this selection, many primary sources are available at the Princeton University Law School Avalon Project and at the Library of Congress Digital Collections (previously LOC webpage, American Memory). Original editions for titles related to the American Revolutionary War can be found open-sourced online at Internet Archive and Hathi Trust Digital Library.
Emmerich, Adreas. The Partisan in War, a treatise on light infantry tactics written by Colonel Andreas Emmerich in 1789.
External links
Maps of the Revolutionary War from the United States Military Academy
Bibliographies online
Library of Congress Guide to the American Revolution
Bibliographies of the War of American Independence compiled by the United States Army Center of Military History
Political bibliography from Omohundro Institute of Early American History and Culture
Conflicts in 1775
Conflicts in 1776
Conflicts in 1777
Conflicts in 1778
Conflicts in 1779
Conflicts in 1780
Conflicts in 1781
Conflicts in 1782
Conflicts in 1783
Global conflicts
Rebellions against the British Empire
Wars between the United Kingdom and the United States
Wars of independence | [
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772_0 | The ampere (, ; symbol: A), often shortened to amp, is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electromagnetism along with the Danish physicist Hans Christian Ørsted.
The International System of Units defines the ampere in terms of other base units by measuring the electromagnetic force between electrical conductors carrying electric current. The earlier CGS system had two definitions of current, one essentially the same as the SI's and the other using electric charge as the base unit, with the unit of charge defined by measuring the force between two charged metal plates. The ampere was then defined as one coulomb of charge per second. In SI, the unit of charge, the coulomb, is defined as the charge carried by one ampere during one second.
New definitions, in terms of invariant constants of nature, specifically the elementary charge, took effect on 20 May 2019.
Definition
The ampere is defined by taking the fixed numerical value of the elementary charge to be 1.602 176 634 × 10−19 when expressed in the unit C, which is equal to A⋅s, where the second is defined in terms of , the unperturbed ground state hyperfine transition frequency of the caesium-133 atom.
The SI unit of charge, the coulomb, "is the quantity of electricity carried in 1 second by a current of 1 ampere". Conversely, a current of one ampere is one coulomb of charge going past a given point per second:
In general, charge is determined by steady current flowing for a time as .
Constant, instantaneous and average current are expressed in amperes (as in "the charging current is 1.2 A") and the charge accumulated (or passed through a circuit) over a period of time is expressed in coulombs (as in "the battery charge is "). The relation of the ampere (C/s) to the coulomb is the same as that of the watt (J/s) to the joule.
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] |
772_1 | History
The ampere is named for French physicist and mathematician André-Marie Ampère (1775–1836), who studied electromagnetism and laid the foundation of electrodynamics. In recognition of Ampère's contributions to the creation of modern electrical science, an international convention, signed at the 1881 International Exposition of Electricity, established the ampere as a standard unit of electrical measurement for electric current.
The ampere was originally defined as one tenth of the unit of electric current in the centimetre–gram–second system of units. That unit, now known as the abampere, was defined as the amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart. The size of the unit was chosen so that the units derived from it in the MKSA system would be conveniently sized.
The "international ampere" was an early realization of the ampere, defined as the current that would deposit of silver per second from a silver nitrate solution. Later, more accurate measurements revealed that this current is .
Since power is defined as the product of current and voltage, the ampere can alternatively be expressed in terms of the other units using the relationship , and thus 1 A = 1 W/V. Current can be measured by a multimeter, a device that can measure electrical voltage, current, and resistance.
Former definition in the SI
Until 2019, the SI defined the ampere as follows:
The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to newtons per metre of length.
Ampère's force law states that there is an attractive or repulsive force between two parallel wires carrying an electric current. This force is used in the formal definition of the ampere.
The SI unit of charge, the coulomb, was then defined as "the quantity of electricity carried in 1 second by a current of 1 ampere". Conversely, a current of one ampere is one coulomb of charge going past a given point per second:
In general, charge was determined by steady current flowing for a time as . | [
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772_2 | Realisation
The standard ampere is most accurately realised using a Kibble balance, but is in practice maintained via Ohm's law from the units of electromotive force and resistance, the volt and the ohm, since the latter two can be tied to physical phenomena that are relatively easy to reproduce, the Josephson effect and the quantum Hall effect, respectively.
Techniques to establish the realisation of an ampere have a relative uncertainty of approximately a few parts in 10, and involve realisations of the watt, the ohm and the volt.
See also
Ammeter
Ampacity (current-carrying capacity)
Electric current
Electric shock
Hydraulic analogy
Magnetic constant
Orders of magnitude (current)
References
External links
The NIST Reference on Constants, Units, and Uncertainty
NIST Definition of ampere and μ0
SI base units
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775_0 | In mathematics and computer science, an algorithm () is a finite sequence of well-defined instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. By making use of artificial intelligence, algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".
In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.
As an effective method, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.
History
The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithm, were used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in 240 BC in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers. Arabic mathematicians such as al-Kindi in the 9th century used cryptographic algorithms for code-breaking, based on frequency analysis. | [
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775_1 | The word algorithm is derived from the name of the 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī, whose nisba (identifying him as from Khwarazm) was Latinized as Algoritmi (Arabized Persian الخوارزمی c. 780–850).
Muḥammad ibn Mūsā al-Khwārizmī was a mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarazm', a region that was part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, which was translated into Latin during the 12th century. The manuscript starts with the phrase Dixit Algorizmi ('Thus spake Al-Khwarizmi'), where "Algorizmi" was the translator's Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the Algebra. In late medieval Latin, algorismus, English 'algorism', the corruption of his name, simply meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός (arithmos), 'number' (cf. 'arithmetic'), the Latin word was altered to algorithmus, and the corresponding English term 'algorithm' is first attested in the 17th century; the modern sense was introduced in the 19th century.
Indian mathematics was predominantly algorithmic.
Algorithms that are representative of the Indian mathematical tradition range from the ancient Śulbasūtrās to the medieval texts of the Kerala School.
In English, the word algorithm was first used in about 1230 and then by Chaucer in 1391. English adopted the French term, but it was not until the late 19th century that "algorithm" took on the meaning that it has in modern English.
Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins with:
which translates to: | [
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775_2 | which translates to:
The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice (Tali Indorum), or Hindu numerals.
A partial formalization of the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Informal definition
An informal definition could be "a set of rules that precisely defines a sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and (for example) any prescribed bureaucratic procedure
or cook-book recipe.
In general, a program is only an algorithm if it stops eventually—even though infinite loops may sometimes prove desirable.
A prototypical example of an algorithm is the Euclidean algorithm, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart above and as an example in a later section.
offer an informal meaning of the word "algorithm" in the following quotation:
No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ... you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols. | [
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775_3 | An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. For example, an algorithm can be an algebraic equation such as y = m + n (i.e., two arbitrary "input variables" m and n that produce an output y), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in a language understood by "the computer") for a fast, efficient, "good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.
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775_4 | Formalization
Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific order—to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):
Minsky: "But we will also maintain, with Turing ... that any procedure which could "naturally" be called effective, can, in fact, be realized by a (simple) machine. Although this may seem extreme, the arguments ... in its favor are hard to refute".
Gurevich: "… Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine … according to Savage [1987], an algorithm is a computational process defined by a Turing machine".Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of computability theory known as the halting problem.
Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.
For some of these computational processes, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).
Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom"—an idea that is described more formally by flow of control. | [
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775_5 | So far, the discussion on the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value of a variable. It derives from the intuition of "memory" as a scratchpad. An example of such an assignment can be found below.
For some alternate conceptions of what constitutes an algorithm, see functional programming and logic programming.
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in the statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also often used as a way to define or document algorithms.
There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see finite-state machine, state transition table and control table for more), as flowcharts and drakon-charts (see state diagram for more), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see Turing machine for more).
Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
1 High-level description
"...prose to describe an algorithm, ignoring the implementation details. At this level, we do not need to mention how the machine manages its tape or head."
2 Implementation description
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level, we do not give details of states or transition function."
3 Formal description
Most detailed, "lowest level", gives the Turing machine's "state table".
For an example of the simple algorithm "Add m+n" described in all three levels, see Examples.
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775_6 | Design
Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming and divide-and-conquer. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern.
One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g. an algorithm's run-time growth as the size of its input increases.
Typical steps in the development of algorithms:
Problem definition
Development of a model
Specification of the algorithm
Designing an algorithm
Checking the correctness of the algorithm
Analysis of algorithm
Implementation of algorithm
Program testing
Documentation preparation
Computer algorithms
"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: " ... we want good algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc."
Chaitin: " ... a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the Halting problem (ibid).
Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".
Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below. | [
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775_7 | Computers (and computors), models of computation: A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability". Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional. | [
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775_8 | Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).
But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement").
Structured programming, canonical structures: Per the Church–Turing thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction. | [
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775_9 | Canonical flowchart symbols: The graphical aide called a flowchart, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols, and their use to build the canonical structures are shown in the diagram.
Examples
Algorithm example
One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:
High-level description:
If there are no numbers in the set then there is no highest number.
Assume the first number in the set is the largest number in the set.
For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set.
When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.
(Quasi-)formal description:
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code:
Input: A list of numbers L.
Output: The largest number in the list L.
if L.size = 0 return null
largest ← L[0]
for each item in L, do
if item > largest, then
largest ← item
return largest | [
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775_10 | Euclid's algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.
For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be "proper"; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero).
Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm. | [
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] |
775_11 | Computer language for Euclid's algorithm
Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction.
A location is symbolized by upper case letter(s), e.g. S, A, etc.
The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009.
An inelegant program for Euclid's algorithm
The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4:
INPUT:
[Into two locations L and S put the numbers l and s that represent the two lengths]:
INPUT L, S
[Initialize R: make the remaining length r equal to the starting/initial/input length l]:
R ← L
E0: [Ensure r ≥ s.]
[Ensure the smaller of the two numbers is in S and the larger in R]:
IF R > S THEN
the contents of L is the larger number so skip over the exchange-steps 4, 5 and 6:
GOTO step 7
ELSE
swap the contents of R and S.
L ← R (this first step is redundant, but is useful for later discussion).
R ← S
S ← L
E1: [Find remainder]: Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R.
IF S > R THEN
done measuring so
GOTO 10
ELSE
measure again,
R ← R − S
[Remainder-loop]:
GOTO 7. | [
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] |
775_12 | E2: [Is the remainder zero?]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S.
IF R = 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO step 11,
E3: [Interchange s and r]: The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location.
L ← R
R ← S
S ← L
[Repeat the measuring process]:
GOTO 7
OUTPUT:
[Done. S contains the greatest common divisor]:
PRINT S
DONE:
HALT, END, STOP.
An elegant program for Euclid's algorithm
The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction LET [] = [] is the assignment instruction symbolized by ←.
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
How "Elegant" works: In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ← A − B, and a B ≤ A loop that computes B ← B − A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend − Subtrahend), the minuend can become s (the new measuring length) and the subtrahend can become the new r (the length to be measured); in other words the "sense" of the subtraction reverses. | [
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775_13 | The following version can be used with programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B){
A=abs(A);
B=abs(B);
while (B!=0){
while (A>B) A=A-B;
B=B-A;
}
return A;
}
Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950.
But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure (June 4, 1996).
Proof of program correctness by use of mathematical induction: Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof. | [
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775_14 | Measuring and improving the Euclid algorithms
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, on average much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed.
Can the algorithms be improved?: Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved?
The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search (examples to be found at Busy beaver), trial and error, cleverness, insight, application of inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.
Algorithmic analysis | [
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] |
775_15 | Algorithmic analysis
It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm which adds up the elements of a list of n numbers would have a time requirement of O(n), using big O notation. At all times the algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. Therefore, it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O(n) if it is counted.
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost O(log n)) outperforms a sequential search (cost O(n) ) when used for table lookups on sorted lists or arrays.
Formal versus empirical
The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.
Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.
Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
Execution efficiency | [
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775_16 | Execution efficiency
To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
Classification
There are various ways to classify algorithms, each with its own merits.
By implementation
One way to classify algorithms is by implementation means. | [
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775_17 | Recursion
A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of Hanoi is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
Logical
An algorithm may be viewed as controlled logical deduction. This notion may be expressed as: Algorithm = logic + control. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming paradigm. In pure logic programming languages, the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant semantics: a change in the axioms produces a well-defined change in the algorithm.
Serial, parallel or distributed
Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a computer network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms and are called inherently serial problems.
Deterministic or non-deterministic
Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics.
Exact or approximate | [
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775_18 | Exact or approximate
While many algorithms reach an exact solution, approximation algorithms seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value.
Quantum algorithm
They run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement. | [
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] |
775_19 | By design paradigm
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are: | [
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775_20 | Brute-force or exhaustive search
This is the naive method of trying every possible solution to see which is best.
Divide and conquer
A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a decrease and conquer algorithm, which solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm.
Search and enumeration
Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking.
Randomized algorithm
Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some randomness. Whether randomized algorithms with polynomial time complexity can be the fastest algorithms for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:
Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time.
Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
Reduction of complexity | [
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775_21 | Reduction of complexity
This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as transform and conquer.
Back tracking
In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution. | [
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] |
775_22 | Optimization problems
For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: | [
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775_23 | Linear programming
When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.
Dynamic programming
When a problem shows optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
The greedy method | [
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775_24 | The greedy method
A greedy algorithm is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization problem.
The heuristic method
In optimization problems, heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include local search, tabu search, simulated annealing, and genetic algorithms. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an approximation algorithm. | [
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775_25 | By field of study
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.
Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.
By complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size:
Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an array element.
Logarithmic time: if the time is a logarithmic function of the input size. E.g. binary search algorithm.
Linear time: if the time is proportional to the input size. E.g. the traverse of a list.
Polynomial time: if the time is a power of the input size. E.g. the bubble sort algorithm has quadratic time complexity.
Exponential time: if the time is an exponential function of the input size. E.g. Brute-force search.
Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.
Continuous algorithms
The adjective "continuous" when applied to the word "algorithm" can mean:
An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis; or
An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer.
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775_26 | Legal issues
Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent.
Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).
History: Development of the notion of "algorithm"
Ancient Near East
The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A Sumerian clay tablet found in Shuruppak near Baghdad and dated to circa 2500 BC described the earliest division algorithm. During the Hammurabi dynasty circa 1800-1600 BC, Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.
Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus circa 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).
Discrete and distinguishable symbols
Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post–Turing machine computations. | [
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] |
775_27 | Manipulation of symbols as "place holders" for numbers: algebra
Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book Al-jabr. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):
Cryptographic algorithms
The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.
Mechanical contrivances with discrete states
The clock: Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"—the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace, mid-19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computer—Babbage's analytical engine, the first device considered a real Turing-complete computer instead of just a calculator—and is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime. | [
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775_28 | Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc.] ...". With this machine he could analyze a "syllogism or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine". | [
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775_29 | Jacquard loom, Hollerith punch cards, telegraphy and telephony – the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid-19th century the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 19th century the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter (ca. 1910) with its punched-paper use of Baudot code on tape.
Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".
Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):
It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned."
Mathematics during the 19th century up to the mid-20th century
Symbols and rules: In rapid succession, the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language". | [
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775_30 | But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules". The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913).
The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers. | [
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775_31 | Effective calculability: In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
Emil Post (1936) and Alan Turing (1936–37, 1939)
Emil Post (1936) described the actions of a "computer" (human being) as follows:
"...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
His symbol space would be
"a two-way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke. | [
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775_32 | "One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise, the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes...
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine
Alan Turing's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences.
Turing—his model of computation is now called a Turing machine—begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.
"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite...
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite... | [
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775_33 | "Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided."
Turing's reduction yields the following:
"The simple operations must therefore include:
"(a) Changes of the symbol on one of the observed squares
"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must, therefore, be taken to be one of the following:
"(A) A possible change (a) of symbol together with a possible change of state of mind.
"(B) A possible change (b) of observed squares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer."
A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing, and Post] ... We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability ... .
"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions". | [
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775_34 | J.B. Rosser (1939) and S.C. Kleene (1943)
J. Barkley Rosser defined an 'effective [mathematical] method' in the following manner (italicization added):
"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–226)
Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion, in particular, Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–37) in their mechanism-models of computation.
Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis. But he did this in the following context (boldface in original):
"12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" (Kleene 1943:273) | [
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] |
775_35 | History after 1950
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially the Church–Turing thesis) and philosophy of mind (especially arguments about artificial intelligence). For more, see Algorithm characterizations.
See also
Abstract machine
Algorithm engineering
Algorithm characterizations
Algorithmic bias
Algorithmic composition
Algorithmic entities
Algorithmic synthesis
Algorithmic technique
Algorithmic topology
Garbage in, garbage out
Introduction to Algorithms (textbook)
List of algorithms
List of algorithm general topics
List of important publications in theoretical computer science – Algorithms
Regulation of algorithms
Theory of computation
Computability theory
Computational complexity theory
Notes
Bibliography | [
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] |
775_36 | Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. .
Includes an excellent bibliography of 56 references.
,
: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
Campagnolo, M.L., Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. 91–109
Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (The Undecidable) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
Davis gives commentary before each article. Papers of Gödel, Alonzo Church, Turing, Rosser, Kleene, and Emil Post are included; those cited in the article are listed here by author's name.
Davis offers concise biographies of Leibniz, Boole, Frege, Cantor, Hilbert, Gödel and Turing with von Neumann as the show-stealing villain. Very brief bios of Joseph-Marie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc.
,
Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.
, 3rd edition 1976[?], (pbk.)
, . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof. | [
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775_37 | Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result).
Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church thesis).
Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
A.A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .]
Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines. | [
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775_38 | Reprinted in The Undecidable, pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called Church–Turing thesis.
Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, The Undecidable)
Cf. in particular the first chapter titled: Algorithms, Turing Machines, and Programs. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an algorithm" (p. 4).
. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
Reprinted in The Undecidable, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms: 2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006 | [
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] |
775_39 | Further reading
Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms. Stanford, California: Center for the Study of Language and Information.
Knuth, Donald E. (2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information.
External links
Dictionary of Algorithms and Data Structures – National Institute of Standards and Technology
Algorithm repositories
The Stony Brook Algorithm Repository – State University of New York at Stony Brook
Collected Algorithms of the ACM – Association for Computing Machinery
The Stanford GraphBase – Stanford University
Articles with example pseudocode
Mathematical logic
Theoretical computer science | [
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777_0 | An annual plant is a plant that completes its life cycle, from germination to the production of seeds, within one growing season, and then dies. The length of growing seasons and period in which they take place vary according to geographical location, and may not correspond to the four traditional seasonal divisions of the year. With respect to the traditional seasons, annual plants are generally categorized into summer annuals and winter annuals. Summer annuals germinate during spring or early summer and mature by autumn of the same year. Winter annuals germinate during the autumn and mature during the spring or summer of the following calendar year.
One seed-to-seed life cycle for an annual can occur in as little as a month in some species, though most last several months. Oilseed rapa can go from seed-to-seed in about five weeks under a bank of fluorescent lamps. This style of growing is often used in classrooms for education. Many desert annuals are therophytes, because their seed-to-seed life cycle is only weeks and they spend most of the year as seeds to survive dry conditions.
Cultivation
In cultivation, many food plants are, or are grown as, annuals, including virtually all domesticated grains. Some perennials and biennials are grown in gardens as annuals for convenience, particularly if they are not considered cold hardy for the local climate. Carrot, celery and parsley are true biennials that are usually grown as annual crops for their edible roots, petioles and leaves, respectively. Tomato, sweet potato and bell pepper are tender perennials usually grown as annuals. Ornamental perennials commonly grown as annuals are impatiens, mirabilis, wax begonia, snapdragon, pelargonium, coleus and petunia. Examples of true annuals include corn, wheat, rice, lettuce, peas, watermelon, beans, zinnia and marigold.
Summer
Summer annuals sprout, flower, produce seed, and die, during the warmer months of the year.
The lawn weed crabgrass is a summer annual.
Winter
Winter annuals germinate in autumn or winter, live through the winter, then bloom in winter or spring. | [
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777_1 | The plants grow and bloom during the cool season when most other plants are dormant or other annuals are in seed form waiting for warmer weather to germinate. Winter annuals die after flowering and setting seed. The seeds germinate in the autumn or winter when the soil temperature is cool.
Winter annuals typically grow low to the ground, where they are usually sheltered from the coldest nights by snow cover, and make use of warm periods in winter for growth when the snow melts. Some common winter annuals include henbit, deadnettle, chickweed, and winter cress.
Winter annuals are important ecologically, as they provide vegetative cover that prevents soil erosion during winter and early spring when no other cover exists and they provide fresh vegetation for animals and birds that feed on them. Although they are often considered to be weeds in gardens, this viewpoint is not always necessary, as most of them die when the soil temperature warms up again in early to late spring when other plants are still dormant and have not yet leafed out.
Even though they do not compete directly with cultivated plants, sometimes winter annuals are considered a pest in commercial agriculture, because they can be hosts for insect pests or fungal diseases (such as ovary smut, Microbotryum sp.) which attack crops being cultivated. The property that they prevent the soil from drying out can also be problematic for commercial agriculture.
Molecular genetics
In 2008, it was discovered that the inactivation of only two genes in one species of annual plant leads to its conversion into a perennial plant. Researchers deactivated the SOC1 and FUL genes (which control flowering time) of Arabidopsis thaliana. This switch established phenotypes common in perennial plants, such as wood formation.
See also
References
External links
Garden plants | [
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779_0 | The anthophytes are a grouping of plant taxa bearing flower-like reproductive structures. They were formerly thought to be a clade comprising plants bearing flower-like structures. The group contained the angiosperms - the extant flowering plants, such as roses and grasses - as well as the Gnetales and the extinct Bennettitales.
Detailed morphological and molecular studies have shown that the group is not actually monophyletic, with proposed floral homologies of the gnetophytes and the angiosperms having evolved in parallel. This makes it easier to reconcile molecular clock data that suggests that the angiosperms diverged from the gymnosperms around 320-300 mya.
Some more recent studies have used the word anthophyte to describe a group which includes the angiosperms and a variety of fossils (glossopterids, Pentoxylon, Bennettitales, and Caytonia), but not the Gnetales.
References
Historically recognized plant taxa | [
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780_0 | An atlas is a collection of maps, originally named after the Ancient Greek deity.
Atlas may also refer to:
Mythology
Atlas (mythology), an Ancient Greek Titanic deity who held up the celestial sphere
Atlas, the first legendary king of Atlantis and further variant of the mythical Titan
Atlas of Mauretania, a legendary king of Mauretania and variant of the mythical Titan
Places
United States
Atlas, California
Atlas, Illinois
Atlas, Texas
Atlas, West Virginia
Atlas, Wisconsin
Atlas District, an area in Washington, D.C.
Atlas Peak AVA, a California wine region
Atlas Township, Michigan
Other
Atlas Cinema, a historic movie theatre in Istanbul, Turkey
Atlas Mountains, a set of mountain ranges in northwestern Africa
Atlas, Nilüfer, a village in Nilüfer district of Bursa Province, Turkey
People with the name
Atlas (graffiti artist), American graffiti artist
Atlas DaBone, American wrestler and football player
Charles Atlas (1892–1972), Italian-American bodybuilder
Charles Atlas (artist)
David Atlas (born 1924), American meteorologist who pioneered weather radar
James Atlas (1949-2019), American writer, editor and publisher
Meir Atlas (1848–1926), Lithuanian rabbi
Natacha Atlas (born 1964), Belgian singer
Nava Atlas, American book artist and author
Omar Atlas (born 1938), former Venezuelan professional wrestler
Scott Atlas (born 1955), American conservative health care policy advisor
Teddy Atlas (born 1956), American boxing trainer and commentator
Tony Atlas (born 1954), American wrestler and bodybuilder
Arts, entertainment, and media
Comics
Atlas (Drawn and Quarterly), a comic book series by Dylan Horrocks
Agents of Atlas, a Marvel Comics mini-series
Atlas Comics (1950s), a publisher
Atlas/Seaboard Comics, a 1970s line of comics
Fictional characters
Atlas (DC Comics), the name of several of DC Comics' fictional characters, comic book superheroes, and deities
Atlas (Teen Titans), Teen Titans character
Atlas, an Astro Boy character
Atlas, a BioShock character
Atlas, a BattleMech in the BattleTech universe
Atlas, an antagonist in Mega Man ZX Advent
Atlas, a Portal 2 character
Atlas, a PS238 character
Erik Josten, a.k.a. Atlas, a Marvel Comics supervillain
The Atlas, a strong driving force from No Man's Sky | [
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] |
780_1 | Literature
Atlas, a photography book by Gerhard Richter
ATLAS of Finite Groups, a group theory book
Atlas Shrugged, a novel by Ayn Rand
The Atlas (novel), by William T. Vollmann
Music
Groups
Atlas (band), a New Zealand rock band
Atlas Sound, the solo musical project of Bradford Cox, lead singer and guitarist of the indie rock band Deerhunter
Musicians
Black Atlass, a Canadian musician
Albums
Atlas (Kinky album)
Atlas (Parkway Drive album), Parkway Drive's fourth album
Atlas (Real Estate album)
Atlas (RÜFÜS album)
Operas
Atlas (opera), 1991 opera by Meredith Monk
Atlas: An Opera in Three Parts, 1993 recording of Monk's opera
Songs
"Atlas" (Battles song), 2007 song by Battles on the album Mirrored
"Atlas" (Coldplay song), 2013 song by Coldplay from The Hunger Games: Catching Fire soundtrack
"Atlas", a song by Caligula's Horse from the album The Tide, the Thief & River's End
"Atlas", the titular song from Parkway Drive's fourth album
"Atlas", a song by Man Overboard from Man Overboard
"Atlas", a song by Jake Chudnow used as main theme in the YouTube series Mind Field
Periodicals
Atlas (magazine)
The Atlas, a newspaper published in England from 1826 to 1869
Other uses in arts, entertainment, and media
Atlas (film)
Atlas (statue), iconic statue by Lee Lawrie in Rockefeller Center
Atlas, a book about flora and/or fauna of a region, such as atlases of the flora and fauna of Britain and Ireland
Atlas Entertainment, a film production company
Atlas folio, a book size
Atlas Media Corp., a non-fiction entertainment company
Atlas Press, a UK publisher
RTV Atlas, a broadcaster in Montenegro
Atlas Sound, a solo musical project by Bradford Cox
The Atlas (video game), a 1991 multiplatform strategy video game
Atlas (video game), an upcoming massively-multiplayer online video game
Atlas Corporation, a fictional arms manufacturer in the video game series Borderlands (series) | [
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780_2 | Brands and enterprises
Atlas (appliance company), a Belarusian company
Atlas Consortium, a group of technology companies
Atlas Copco, Swedish company founded in 1873
Atlas Corporation, an investment company
Atlas Elektronik, a German naval/marine electronics and systems business
Atlas Group, a Pakistani business group
Atlas Mara Limited, formerly Atlas Mara Co-Nvest Limited, a financial holding company that owns banks in Africa
Atlas Model Railroad, American maker of model trains and accessories
Atlas Network, formerly Atlas Economic Research Foundation
Atlas Press (tool company)
Atlas Solutions, a subsidiary of Facebook for digital online advertising, formerly owned by Microsoft
Atlas Telecom, a worldwide communications company
Atlas Van Lines, a moving company
Atlas-Imperial, an American diesel engine manufacturer
Dresser Atlas, a provider of oilfield and factory automation services
Tele Atlas, a Dutch mapping company
Western Atlas, an oilfield services company
Computing and technology
Atlas (computer), an early supercomputer, built in the 1960s
Atlas (robot), a humanoid robot developed by Boston Dynamics and DARPA
ATLAS (software), a software flagging naturalized American for denaturalization
Atlas, a computer used at the Lawrence Livermore National Laboratory in 2006
Abbreviated Test Language for All Systems, or ATLAS, a MILSPEC language for avionics equipment testing
Advanced Technology Leisure Application Simulator, or ATLAS, a hydraulic motion simulator used in theme parks
ASP.NET AJAX (formerly "Atlas"), a set of ASP.NET extensions
ATLAS Transformation Language, programming language
Atlas.ti, a qualitative analysis program
Automatically Tuned Linear Algebra Software, or ATLAS,
Texture atlas, or image sprite sheet
UNIVAC 1101, an early American computer, built in the 1950s
Science
Astronomy
Atlas (comet) (C/2019 Y4)
Atlas (crater) on the near side of the Moon
Atlas (moon), a satellite of Saturn
Atlas (star), also designated 27 Tauri, a triple star system in the constellation of Taurus and a member of the Pleiades
Advanced Technology Large-Aperture Space Telescope (ATLAST)
Advanced Topographic Laser Altimeter System (ATLAS), a space-based lidar instrument on ICESat-2
Asteroid Terrestrial-impact Last Alert System (ATLAS)
Mathematics
Atlas (manifolds), a set of smooth charts
Atlas (topology), a set of charts
Smooth atlas | [
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] |
780_3 | Physics
Argonne Tandem Linear Accelerator System, or ATLAS, a linear accelerator at the Argonne National Laboratory
ATLAS experiment, a particle detector for the Large Hadron Collider at CERN
Atomic-terrace low-angle shadowing, or ATLAS, a nanofabrication technique
Biology and healthcare
Atlas (anatomy), part of the spine
Atlas personality, a term used in psychology to describe the personality of someone whose childhood was characterized by excessive responsibilities
Brain atlas, a neuroanatomical map of the brain of a human or other animal
Animals and plants
Atlas bear
Atlas beetle
Atlas cedar
Atlas moth
Atlas pied flycatcher, a bird
Atlas turtle
Sport
Atlas Delmenhorst, a German association football club
Atlas F.C., a Mexican professional football club
Club Atlético Atlas, an Argentine amateur football club
KK Atlas, a former men's professional basketball club based in Belgrade (today's Serbia)
Transport
Aerospace
Atlas (rocket family)
SM-65 Atlas intercontinental ballistic missile (ICBM)
AeroVelo Atlas, a human-powered helicopter
Airbus A400M Atlas, a military aircraft produced 2007–present
Armstrong Whitworth Atlas, a British military aeroplane produced 1927–1933
Atlas Air, an American cargo airline
Atlas Aircraft, a 1940s aircraft manufacturer
Atlas Aircraft Corporation, a South African military aircraft manufacturer
Atlas Aviation, an aircraft maintenance firm
Atlas Blue, a Moroccan low-cost airline
Atlasjet, a Turkish airline
Birdman Atlas, an ultralight aircraft
HMLAT-303, U.S. Marine Corps helicopter training squadron
La Mouette Atlas, a French hang glider design
Automotive
Atlas (1951 automobile), a French mini-car
Atlas (light trucks), a Greek motor vehicle manufacturer
Atlas (Pittsburgh automobile), produced 1906–1907
Atlas (Springfield automobile), produced 1907–1913
Atlas, a British van by the Standard Motor Company produced 1958–1962
Atlas Drop Forge Company, a parts subsidiary of REO Motor Car Company
Atlas Motor Buggy, an American highwheeler produced in 1909
General Motors Atlas engine
Honda Atlas Cars Pakistan, a Pakistani car manufacturer
Nissan Atlas, a Japanese light truck
Volkswagen Atlas, a sport utility vehicle
Geely Atlas, a sport utility vehicle
Ships and boats
Atlas Werke, a former German shipbuilding company
, the name of several Royal Navy ships
ST Atlas, a Swedish tugboat
, the name of several U.S. Navy ships | [
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780_4 | Trains
Atlas, an 1863–1885 South Devon Railway Dido class locomotive
Atlas, a 1927–1962 LMS Royal Scot Class locomotive
Atlas Car and Manufacturing Company, a locomotive manufacturer
Atlas Model Railroad
Other uses
Atlas (architecture)
ATLAS (simulation) (Army Tactical Level Advanced Simulation), a Thai military system
Atlas (storm), which hit the Midwestern United States in October 2013, named by The Weather Channel
Agrupación de Trabajadores Latinoamericanos Sindicalistas, or ATLAS, a former Latin American trade union confederation in the early 1950s
Atlas languages, Berber languages spoken in the Atlas Mountains of Morocco
ATLAS Network, a network of European special police units
Atlas Uranium Mill
See also
Altas (disambiguation) | [
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782_0 | Mouthwash, mouth rinse, oral rinse, or mouth bath is a liquid which is held in the mouth passively or swilled around the mouth by contraction of the perioral muscles and/or movement of the head, and may be gargled, where the head is tilted back and the liquid bubbled at the back of the mouth.
Usually mouthwashes are antiseptic solutions intended to reduce the microbial load in the mouth, although other mouthwashes might be given for other reasons such as for their analgesic, anti-inflammatory or anti-fungal action. Additionally, some rinses act as saliva substitutes to neutralize acid and keep the mouth moist in xerostomia (dry mouth). Cosmetic mouthrinses temporarily control or reduce bad breath and leave the mouth with a pleasant taste.
Rinsing with water or mouthwash after brushing with a fluoride toothpaste can reduce the availability of salivary fluoride. This can lower the anti-cavity re-mineralization and antibacterial effects of fluoride. Fluoridated mouthwash may mitigate this effect or in high concentrations increase available fluoride, but is not as cost effective as leaving the fluoride toothpaste on the teeth after brushing. A group of experts discussing post brushing rinsing in 2012 found that although there was clear guidance given in many public health advice publications to "spit, avoid rinsing with water/excessive rinsing with water" they believed there was a limited evidence base for best practice.
Use
Common use involves rinsing the mouth with about 20-50 ml (2/3 fl oz) of mouthwash. The wash is typically swished or gargled for about half a minute and then spat out. Most companies suggest not drinking water immediately after using mouthwash. In some brands, the expectorate is stained, so that one can see the bacteria and debris.
Mouthwash should not be used immediately after brushing the teeth so as not to wash away the beneficial fluoride residue left from the toothpaste. Similarly, the mouth should not be rinsed out with water after brushing. Patients were told to "spit don't rinse" after toothbrushing as part of a National Health Service campaign in the UK. A fluoride mouthrinse can be used at a different time of the day to brushing. | [
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782_1 | Gargling is where the head is tilted back, allowing the mouthwash to sit in the back of the mouth while exhaling, causing the liquid to bubble. Gargling is practiced in Japan for perceived prevention of viral infection. One commonly used way is with infusions or tea. In some cultures, gargling is usually done in private, typically in a bathroom at a sink so the liquid can be rinsed away.
Effects
The most-commonly-used mouthwashes are commercial antiseptics, which are used at home as part of an oral hygiene routine. Mouthwashes combine ingredients to treat a variety of oral conditions. Variations are common, and mouthwash has no standard formulation, so its use and recommendation involves concerns about patient safety. Some manufacturers of mouthwash state that their antiseptic and antiplaque mouthwashes kill the bacterial plaque that causes cavities, gingivitis, and bad breath. It is, however, generally agreed that the use of mouthwash does not eliminate the need for both brushing and flossing. The American Dental Association asserts that regular brushing and proper flossing are enough in most cases, in addition to regular dental check-ups, although they approve many mouthwashes.
For many patients, however, the mechanical methods could be tedious and time-consuming, and, additionally, some local conditions may render them especially difficult. Chemotherapeutic agents, including mouthwashes, could have a key role as adjuncts to daily home care, preventing and controlling supragingival plaque, gingivitis and oral malodor.
Minor and transient side effects of mouthwashes are very common, such as taste disturbance, tooth staining, sensation of a dry mouth, etc. Alcohol-containing mouthwashes may make dry mouth and halitosis worse, as they dry out the mouth. Soreness, ulceration and redness may sometimes occur (e.g., aphthous stomatitis or allergic contact stomatitis) if the person is allergic or sensitive to mouthwash ingredients, such as preservatives, coloring, flavors and fragrances. Such effects might be reduced or eliminated by diluting the mouthwash with water, using a different mouthwash (e.g. saltwater), or foregoing mouthwash entirely. | [
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782_2 | Prescription mouthwashes are used prior to and after oral surgery procedures, such as tooth extraction, or to treat the pain associated with mucositis caused by radiation therapy or chemotherapy. They are also prescribed for aphthous ulcers, other oral ulcers, and other mouth pain. "Magic mouthwashes" are prescription mouthwashes compounded in a pharmacy from a list of ingredients specified by a doctor. Despite a lack of evidence that prescription mouthwashes are more effective in decreasing the pain of oral lesions, many patients and prescribers continue to use them. There has been only one controlled study to evaluate the efficacy of magic mouthwash; it shows no difference in efficacy between the most common magic-mouthwash formulation, on the one hand, and commercial mouthwashes (such as chlorhexidine) or a saline/baking soda solution, on the other. Current guidelines suggest that saline solution is just as effective as magic mouthwash in pain relief and in shortening the healing time of oral mucositis from cancer therapies.
History
The first known references to mouth rinsing is in Ayurveda for treatment of gingivitis. Later, in the Greek and Roman periods, mouth rinsing following mechanical cleansing became common among the upper classes, and Hippocrates recommended a mixture of salt, alum, and vinegar. The Jewish Talmud, dating back about 1,800 years, suggests a cure for gum ailments containing "dough water" and olive oil.
Before Europeans came to the Americas, Native North American and Mesoamerican cultures used mouthwashes, often made from plants such as Coptis trifolia. Indeed, Aztec dentistry was more advanced than European dentistry of the age. Peoples of the Americas used salt water mouthwashes for sore throats, and other mouthwashes for problems such as teething and mouth ulcers. | [
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782_3 | Anton van Leeuwenhoek, the famous 17th century microscopist, discovered living organisms (living, because they were mobile) in deposits on the teeth (what we now call dental plaque). He also found organisms in water from the canal next to his home in Delft. He experimented with samples by adding vinegar or brandy and found that this resulted in the immediate immobilization or killing of the organisms suspended in water. Next he tried rinsing the mouth of himself and somebody else with a mouthwash containing vinegar or brandy and found that living organisms remained in the dental plaque. He concluded—correctly—that the mouthwash either did not reach, or was not present long enough, to kill the plaque organisms.
In 1892, German Richard Seifert invented mouthwash product Odol, which was produced by company founder Karl August Lingner (1861–1916) in Dresden.
That remained the state of affairs until the late 1960s when Harald Loe (at the time a professor at the Royal Dental College in Aarhus, Denmark) demonstrated that a chlorhexidine compound could prevent the build-up of dental plaque. The reason for chlorhexidine's effectiveness is that it strongly adheres to surfaces in the mouth and thus remains present in effective concentrations for many hours.
Since then commercial interest in mouthwashes has been intense and several newer products claim effectiveness in reducing the build-up in dental plaque and the associated severity of gingivitis, in addition to fighting bad breath. Many of these solutions aim to control the Volatile Sulfur Compound (VSC)-creating anaerobic bacteria that live in the mouth and excrete substances that lead to bad breath and unpleasant mouth taste. For example, the number of mouthwash variants in the United States of America has grown from 15 (1970) to 66 (1998) to 113 (2012).
Research
Research in the field of microbiotas shows that only a limited set of microbes cause tooth decay, with most of the bacteria in the human mouth being harmless. Focused attention on cavity-causing bacteria such as Streptococcus mutans has led research into new mouthwash treatments that prevent these bacteria from initially growing. While current mouthwash treatments must be used with a degree of frequency to prevent this bacteria from regrowing, future treatments could provide a viable long-term solution.
Ingredients
Alcohol | [
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782_4 | Ingredients
Alcohol
Alcohol is added to mouthwash not to destroy bacteria but to act as a carrier agent for essential active ingredients such as menthol, eucalyptol and thymol, which help to penetrate plaque. Sometimes a significant amount of alcohol (up to 27% vol) is added, as a carrier for the flavor, to provide "bite". Because of the alcohol content, it is possible to fail a breathalyzer test after rinsing, although breath alcohol levels return to normal after 10 minutes. In addition, alcohol is a drying agent, which encourages bacterial activity in the mouth, releasing more malodorous volatile sulfur compounds. Therefore, alcohol-containing mouthwash may temporarily worsen halitosis in those who already have it, or, indeed, be the sole cause of halitosis in other individuals.
It is hypothesized that alcohol in mouthwashes acts as a carcinogen (cancer-inducing agent). Generally, there is no scientific consensus about this. One review stated:
The same researchers also state that the risk of acquiring oral cancer rises almost five times for users of alcohol-containing mouthwash who neither smoke nor drink (with a higher rate of increase for those who do). In addition, the authors highlight side effects from several mainstream mouthwashes that included dental erosion and accidental poisoning of children. The review garnered media attention and conflicting opinions from other researchers. Yinka Ebo of Cancer Research UK disputed the findings, concluding that "there is still not enough evidence to suggest that using mouthwash that contains alcohol will increase the risk of mouth cancer". Studies conducted in 1985, 1995, 2003, and 2012 did not support an association between alcohol-containing mouth rinses and oral cancer. Andrew Penman, chief executive of The Cancer Council New South Wales, called for further research on the matter. In a March 2009 brief, the American Dental Association said "the available evidence does not support a connection between oral cancer and alcohol-containing mouthrinse". Many newer brands of mouthwash are alcohol free, not just in response to consumer concerns about oral cancer, but also to cater for religious groups who abstain from alcohol consumption. | [
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782_5 | Benzydamine (analgesic)
In painful oral conditions such as aphthous stomatitis, analgesic mouthrinses (e.g. benzydamine mouthwash, or "Difflam") are sometimes used to ease pain, commonly used before meals to reduce discomfort while eating.
Benzoic acid
Benzoic acid acts as a buffer.
Betamethasone
Betamethasone is sometimes used as an anti-inflammatory, corticosteroid mouthwash. It may be used for severe inflammatory conditions of the oral mucosa such as the severe forms of aphthous stomatitis.
Cetylpyridinium chloride (antiseptic, antimalodor)
Cetylpyridinium chloride containing mouthwash (e.g. 0.05%) is used in some specialized mouthwashes for halitosis. Cetylpyridinium chloride mouthwash has less anti-plaque effect than chlorhexidine and may cause staining of teeth, or sometimes an oral burning sensation or ulceration. | [
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782_6 | Chlorhexidine digluconate and hexetidine (antiseptic)
Chlorhexidine digluconate is a chemical antiseptic and is used in a 0.12–0.2% solution as a mouthwash. However, there is no evidence to support that higher concentrations are more effective in controlling dental plaque and gingivitis. It has anti-plaque action, but also some anti-fungal action. It is especially effective against Gram-negative rods. The proportion of Gram-negative rods increase as gingivitis develops, so it is also used to reduce gingivitis. It is sometimes used as an adjunct to prevent dental caries and to treat gingivitis periodontal disease, although it does not penetrate into periodontal pockets well. Chlorhexidine mouthwash alone is unable to prevent plaque, so it is not a substitute for regular toothbrushing and flossing. Instead, chlorhexidine mouthwash is more effective when used as an adjunctive treatment with toothbrushing and flossing. In the short term, if toothbrushing is impossible due to pain, as may occur in primary herpetic gingivostomatitis, chlorhexidine mouthwash is used as a temporary substitute for other oral hygiene measures. It is not suited for use in acute necrotizing ulcerative gingivitis, however. Rinsing with chlorhexidine mouthwash before a tooth extraction reduces the risk of a dry socket, a painful condition where the blood clot is lost from an extraction socket and bone is exposed to the oral cavity. Other uses of chlorhexidine mouthwash include prevention of oral candidiasis in immunocompromised persons, treatment of denture-related stomatitis, mucosal ulceration/erosions and oral mucosal lesions, general burning sensation and many other uses. | [
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782_7 | Chlorhexidine has good substantivity (the ability of a mouthwash to bind to hard and soft tissues in the mouth). However, chlorhexidine binds to tannins, meaning that prolonged use in persons who consume coffee, tea or red wine is associated with extrinsic staining (i.e. removable staining) of teeth. Chlorhexidine mouthwash can also cause taste disturbance or alteration. Chlorhexidine is rarely associated with other issues like overgrowth of enterobacteria in persons with leukemia, desquamation and irritation of oral mucosa, salivary gland pain and swelling, and hypersensitivity reactions including anaphylaxis. A randomized clinical trial conducted in Rabat University in Morocco found better results in plaque inhibition when chlorohexidine with alcohol base 0.12% was used, when compared to an alcohol-free 0.1% chlorhexidine mouthrinse. Chlorhexidine mouthwashes increase staining of teeth over a period of time. However, many publications, and, in recent times, also a systematic review (van Swaaij 2020) revealed that an AntiDiscoloration System (ADS) based on L-ascorbic acid and sodium metabisulfite is able to reduce tooth staining without affecting the antibacterial effect of chlorhexidine.
Hexetidine also has anti-plaque, analgesic, astringent and anti-malodor properties, but is considered an inferior alternative to chlorhexidine.
Edible oils
In traditional Ayurvedic medicine, the use of oil mouthwashes is called "Kavala" ("oil swishing") or "Gandusha", and this practice has more recently been re-marketed by the complementary and alternative medicine industry as "oil pulling". Its promoters claim it works by "pulling out" "toxins", which are known as ama in Ayurvedic medicine, and thereby reducing inflammation. Ayurvedic literature claims that oil pulling is capable of improving oral and systemic health, including a benefit in conditions such as headaches, migraines, diabetes mellitus, asthma, and acne, as well as whitening teeth. | [
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