ALL

Calculus operations

Derivative, integral, slope at a point, local extrema, points of inflection, and critical points

Treaty of Portsmouth

(1905) ended the Russo-Japanese War (1904-1905). It was signed at the [blank] Naval Shipyard in Maine after negotiations brokered by Theodore Roosevelt (for which he won the Nobel Peace Prize). Japan had dominated the war and received an indemnity, the Liaodong Peninsula in Manchuria, and half of Sakhalin Island, but the treaty was widely condemned in Japan because the public had expected more.

Treaty of Tordesillas

(1494) ostensibly divided the New World (and, in later interpretations, the entire world) between Spain and Portugal. It resulted from a bull by (Spanish-born) Pope Alexander VI granting lands to Spain and established a line west of the Cape Verde islands between future Spanish possessions (west) and Portuguese possessions (east). The line passed through Brazil, allowing the Portuguese to establish a colony there, while Spain received the rest of the Americas. Endless wrangling and repeated revisions ensued.

Peace of Westphalia

(1648) the collective name for two treaties ending the Thirty Years' War that were signed by the Holy Roman Empire, minor German states, Spain, France, Sweden, and the Dutch Republic. It confirmed the principle of "cuius regio eius religio" (that a ruler's religion determined that of his country) introduced by the Peace of Augsburg, but mandated relative tolerance of other (Christian) faiths. It adjusted the borders of German states and strengthened their princes with respect to the Emperor, and transferred most of Lorraine and some of Alsace to France.

Treaty of Utrecht

(1713) was a series of treaties signed in the Dutch city of [blank] that (mostly) ended the War of the Spanish Succession (1701-1714). They were signed by France and Spain for one side and by Britain, Savoy, and the United Provinces (The Netherlands) for the other. The treaty confirmed a Bourbon prince (Philip, Duke of Anjou) on the Spanish throne, ending Habsburg control, but took steps to prevent the French and Spanish thrones from being merged. Some Spanish possessions — including Sicily, the Spanish Netherlands, Naples, and Gibraltar — were given to the victors.

Treaty of Ghent

(1814) ended the War of 1812 between the U.S. and Britain. It was signed in the Belgian city [blank] but, due to the distances involved, could not prevent the Battle of New Orleans two weeks later. The treaty made no boundary changes and had minimal effect; both sides were ready for peace and considered the war a futile and fruitless endeavor.

Adams-Onís Treaty

(1819) settled a boundary dispute between the U.S. and Spain that arose following the Louisiana Purchase. It was negotiated by then-Secretary of State John Quincy Adams and most notably sold Florida to the U.S. in exchange for the payment of its citizens' claims against Spain. It also delineated the U.S.-Spain border to the Pacific Ocean leading to its alternate name, the Transcontinental Treaty.

Treaty of Guadalupe Hidalgo

(1848) ended the Mexican-American War (1846-1848) and was signed in its namesake neighborhood of Mexico City. Its most significant result was the "Mexican Cession" transferring California, Nevada, Utah, and parts of four other states to the U.S. It also made the Rio Grande the boundary between Texas and Mexico.

Treaty of Brest-Litovsk

(1918) was a "separate peace" signed by the Bolshevik government of the new USSR and Germany. The USSR needed to make peace to focus on defeating the "Whites" (royalists) in the Russian Civil War, and it gave up Ukraine, Belarus, and the three Baltic countries after Germany invaded, an outcome worse than a German offer which chief Soviet negotiator Leon Trotsky had rejected. The treaty was negotiated in [blank] (in what is now Belarus) and was nullified by the subsequent Treaty of Versailles following Germany's defeat.

Treaty of Versailles

(1919) officially ended World War I between the Allies and Germany. It was signed at its namesake French palace after the Paris Peace Conference. It is noted for the "Big Four" (Woodrow Wilson, David Lloyd George, Georges Clemenceau, and Vittorio Orlando) who headed the Allies' delegations, discussions of Woodrow Wilson's Fourteen Points (particularly the League of Nations), and its controversial disarmament, war guilt, and reparations clauses. The conference was also notable for up-and-coming world figures who attended (John Maynard Keynes, Ho Chi Minh, Jan Smuts, etc.).

Lateran Treaty

(1929) created the independent country of the Vatican City, made Catholicism the state religion of Italy (ended in 1984), and determined the proper remuneration for Church property taken by Italy. It was signed by Benito Mussolini and a representative of Pope Pius XI in the namesake papal residence, and ended the so-called "Roman Question" that arose out of the unification of Italy and the dissolution of the Papal States.

Camp David Accords

(1978) were negotiated at the presidential retreat [blank] by Egypt's Anwar Sadat and Israel's Menachem Begin; they were brokered by U.S. President Jimmy Carter. They led to a peace treaty the next year that returned the Sinai Peninsula to Egypt, guaranteed Israeli access to the Red Sea and Suez Canal, and more-or-less normalized diplomatic and economic relations between the two countries. This isolated Egypt from the other Arab countries and led to Sadat's assassination in 1981.

Battle of Okinawa

(April 1945 - June 1945): The largest amphibious assault of the Pacific Theater, the [blank] featured massive casualties among both combatants and civilians. The Japanese launched over 1,500 kamikaze attacks against the U.S. fleet, and even sent the massive battleship Yamato on a suicide mission; it was sunk by aircraft before reaching [blank]. On the American side, both war correspondent Ernie Pyle and Lieutenant General Simon Bolivar Buckner Jr., the commander-in-chief of the ground forces, were killed. Somewhat uniquely, the battle also saw large numbers of Japanese troops surrender, although many were native [blank]ns forced into fighting.

Battle of Stalingrad

(August 1942 - February 1943): With about two million casualties, the [blank] is often cited as the bloodiest battle in history. The battle arose out of Germany's summer campaign to capture vital oil supplies in the Caucasus Mountains, but Friedrich Paulus's 6th Army became bogged down in intense street fighting in the city, allowing Soviet Marshal Georgy Zhukov to launch Operation Uranus, which encircled Paulus's men by defeating the Italian, Hungarian, and Romanian forces guarding their flank. In the final days of the battle, Hitler promoted Paulus to field marshal, a not-so-subtle suggestion that Paulus should either fight to the death or commit suicide, as no German field marshal had ever been captured; Paulus surrendered anyway.

Battle of the Bulge

(December 1944 - January 1945): The [blank] resulted from Germany's last major offensive operation on the Western Front. The German plan to sweep through the Ardennes Forest and capture the port city Antwerp, Belgium, benefited from Allied aircraft being grounded due to poor weather. During the battle, English-speaking German troops under Otto Skorzeny attempted to disguise themselves as Allied troops and infiltrate enemy lines. German forces also besieged the Belgian town Bastogne and requested its surrender, to which U.S. Army Brigadier General Anthony McAuliffe replied "Nuts!"; the siege was eventually lifted by forces commanded by George Patton.

Attack on Pearl Harbor

(December 7, 1941): On what President Franklin Roosevelt declared would be "a date which will live in infamy," Japanese carrier-based aircraft launched — without a formal declaration of war — a surprise attack on the American naval base at [blank] on the Hawaiian island Oahu. The attack sank four battleships, most notably the USS Arizona, but all of the U.S. Navy's carriers were at sea and were unattacked. Shortly after the attack, Japan began invasions of Guam, Wake Island, the Philippines, and the British colony of Singapore. On December 8, with only Montana Representative Jeannette Rankin dissenting, the U.S. Congress declared war on Japan.

Surface area of a parallelepiped

2ab+2bc+2ac

Battle of Iwo Jima

(February 1945 - March 1945): The Allies sought to capture [blank], a small island midway between the Mariana Islands and the Japanese home islands, to provide an airbase for the eventual invasion of Japan. Under the leadership of General Tadamichi Kuribayashi, the island's defenders built a complex network of underground tunnels and well-camouflaged artillery pieces that enabled them to hold out for a month against vastly superior forces. The battle is best known for Joe Rosenthal's photograph showing six American servicemen raising a flag atop Mount Suribachi.

Battle of Britain

(July 1940 - October 1940): [blank] saw the British Royal Air Force (RAF) defeat the German air force, known as the Luftwaffe, effectively saving Britain from a proposed German amphibious invasion codenamed Operation Sea Lion. The primary German fighter plane was the Messerschmitt Bf 109, which engaged in numerous dogfights against British pilots flying Hurricane and Spitfire aircraft. Effective use of radar helped to repel German forces, forcing the Luftwaffe into nighttime raids against civilian targets, a campaign known as the Blitz.

Battle of Kursk

(July 1943 - August 1943): Fought in western Russia, the [blank] was the largest tank battle in history, with about 6,000 tanks engaged. Thanks to a complex spy network, the Soviet leadership was well-informed about German plans to launch Operation Citadel against the Kursk salient, and constructed massive defensive fortifications. After the German advance was stopped, a successful Soviet counterattack was launched. The German Army never again was able to mount a major attack on the Eastern Front.

Battle of Midway

(June 1942): Universally considered the turning point in the Pacific Theater, the [blank] saw the Japanese lose four aircraft carriers, a blow from which they never fully recovered. Japanese Admiral Isoroku Yamamoto planned to lure the U.S. fleet into a trap, but the Americans had broken the Japanese code, allowing them to pull off a stunning victory, with dive bombers from the Enterprise sinking the carriers Kaga, Akagi, and Hiryu, while those from the hastily-repaired Yorktown sank the carrier Soryu.

D-Day (Operation Overlord)

(June 6, 1944): The largest amphibious assault in history, as Supreme Allied Commander Dwight Eisenhower's forces attacked the German Atlantic Wall defenses on the beaches of Normandy, France. Due to his wife's birthday, German Field Marshal Erwin Rommel was absent at the start the invasion, which saw American forces land at Utah and Omaha Beaches, British forces land at Gold and Sword Beaches, and Canadian forces land at Juno Beach. After the landings, Allied forces erected prefabricated artificial Mulberry harbors to aid in transporting goods to France.

Battle of the Coral Sea

(May 1942): Resulting from a Japanese plan to capture Port Moresby, New Guinea, the [blank] was fought entirely by carrier-based aircraft, making it the first major naval battle in history in which the two opposing fleets never directly fired upon (or even sighted) each other. The U.S. Navy's carrier Lexington was sunk, and the Yorktown heavily damaged, while the Japanese navy lost the light carrier Shoho and saw its large carriers Shokaku and Zuikaku damaged. Ultimately, the invasion of Port Moresby was cancelled and the temporary loss of two Japanese carriers gave the U.S. an edge at the subsequent Battle of Midway.

Battle of El Alamein

(October 1942 - November 1942): The Second [blank] marked the turning point in the African campaign. Named for an Egyptian coastal town 65 miles west of Alexandria, it saw the British Eighth Army under Bernard Montgomery defeat the German Afrika Korps under Erwin Rommel, preventing the Nazis from capturing the Suez Canal and oil fields in the Middle East. Following the battle, Allied forces landed in Morocco and Algeria as part of Operation Torch, and by May 1943 all Axis forces in North Africa had surrendered.

Battle of Leyte Gulf

(October 1944): By some measures the largest naval battle in history, the [blank] resulted from the Japanese Sho-Go plan to halt the American reconquest of the Philippines. The plan nearly worked when American Admiral William "Bull" Halsey was baited into moving all of his battleships and large carriers away from the landing site, but an American force of small escort carriers and destroyers held off a Japanese task force that included four battleships. Another Japanese force tried to pass through the Surigao Strait, but — in the last-ever combat between opposing battleships — the American Seventh Fleet crossed their 'T' and annihilated the force.

Area of triangle

1/2 (base)(height)

Surface area of a sphere

4 x pi x r^2

Volume of a sphere

4/3 x pi x r^3

Hastings (1066)

A little more than two weeks after Stamford Bridge, Harold Godwinson was defeated by another claimant to the throne of England at Hastings. That claimant was Duke William of Normandy, who is now known as William the Conqueror. The battle and Harold's death are depicted in the Bayeux Tapestry. The battle took place two weeks after William's landing at Pevensey.

The Pyramids (July 21, 1798, Egypt) and the Nile (August 1-3, 1798, Egypt)

After his victory over Austria, Napoleon proposed crossing the Mediterranean and invading Egypt. While the stated goal of this expedition was to strike a blow against British trade with the Middle East and Asia, it also catered to Napoleon's fascination with antiquity. A team of scientists followed his military expedition, whose most lasting result was the discovery of the Rosetta Stone, unearthed by soldiers digging to construct a fort in the Nile delta. Napoleon supposedly cried "Soldiers, forty centuries look down upon you" at the 1798 Battle of the Pyramids, where his troops used modern artillery and large square formations to ward off a cavalry charge by the Egyptian Mamluks. French control of Egypt, however, was dependent on communications across the Mediterranean, which were interrupted by the Royal Navy's attack on the French fleet in Aboukir Bay in early August. British victory at the Nile forced Napoleon to abandon his army and return to France.

Battle of Okinawa (April to June 1945)

Codenamed Operation Iceberg, the invasion of ____________ was the last major ground battle in the Pacific theater of World War II. Kakazu Ridge and Shuri Castle were among the positions Japanese troops defended against U.S. Marine and Army units. Okinawa was the high point of Japanese kamikaze attacks; over 1,500 suicide missions were sent against the Allied invasion fleet. The heavy casualties sustained by both sides on Okinawa and the suicidal bravery of the Japanese military encouraged American planners to use atomic weapons on Hiroshima and Nagasaki instead of launching Operation Downfall, the projected late-1945 conventional invasion of the Japanese home islands.

Constantinople (1453)

Constantinople fell to the Ottomans on May 29, 1453 after a seven-week siege. The Byzantine defenders were led by Emperor Constantine XI and a Genoese captain named Giovanni Giustiniani, and the Ottomans were led by Sultan Mehmed II and Zaganos Pasha. Preparations for the siege included the building of Rumeli Hisar across from Anadolu Hisar on the Bosphorus. The defenders stretched a chain across the Golden Horn, but the Ottomans rolled their ships over Galata Hill on greased logs to encircle the city.

Permutations

EX: Suppose we want to find the number of ways to arrange the three letters in the word CAT in different two-letter groups where CA is different from AC and there are no repeated letters. - is known as 3_P_2, which is 3!/(3-2)!

Bannockburn (1314)

The Battle of Bannockburn was a decisive victory for Robert the Bruce, King of Scots, over King Edward II of England. The English attempted to lift the siege of Stirling Castle. The battle began with single combat between Robert the Bruce and Sir Henry de Bohun. The Earl of Moray commanded the Scottish vanguard near the Church of St. Ninian.

Battle of Bataan (January to May 1942)

Immediately after the attack on Pearl Harbor, Japanese bombers struck the Philippine island Luzon, forcing the Americans to retreat to the Bataan Peninsula, where they held out for four months. General Douglas MacArthur vowed "I shall return" before evacuating to Australia and leaving command to Jonathan Wainwright, who retreated to the island of Corregidor and surrendered on May 6. The prisoners from Bataan were sent on a "death march" 80 miles to San Fernando with minimal food, water, and medical supplies; those that fell behind were beaten. Japanese General Masaharu Homma was executed in 1946 for his role in the war crime.

Trafalgar (October 21, 1805, off the coast of southwestern Spain)

In 1804 Napoleon abolished the consulate and became France's emperor. He faced an array of enemies who made up the "Third Coalition": the humiliated Austrians sought military aid from both Russia and Britain. France and Spain allied in the hope of challenging the Royal Navy and making it possible for Napoleon's armies to launch an invasion of Britain. Trafalgar, fought in the Atlantic off the coast of Spain in the fall of 1805, was the last great naval battle of the Napoleonic era. A combined French and Spanish fleet under Admiral Pierre-Charles Villeneuve was attacked by Royal Navy ships under Lord Horatio Nelson, Britain's greatest admiral. Just before the battle, Nelson's flagship — the HMS Victory — flew the signal "England expects that every man will do his duty." Although Nelson was killed by a French sniper in the battle that followed, his ships captured half of the French fleet, including Admiral Villeneuve. French plans to invade Britain were postponed indefinitely.

Jena-Auerstedt (October 14, 1806, Germany)

In 1806 Napoleon turned his forces against Prussia. At the twin October battles Jena and Auerstedt, Napoleon and Marshal Louis-Nicolas Davout worked together to smash a Prussian army led by the Duke of Brunswick, who was mortally wounded by Davout's troops near Auerstadt. The collapse of the Prussian army, widely considered the continent's most experienced and professional military force, shocked European observers. Napoleon felt that he had secured revenge for Frederick the Great's victory over France at Rossbach in the Seven Years' War (1757).

Borodino (September 7, 1812, Russia)

In 1812 Napoleon assembled the largest army of his reign for the most ambitious military operation of the 19th century: a full-scale invasion of Russia. Russian Tsar Alexander I, despite losing to Napoleon in central Europe between 1805 and 1807, refused to agree to the "Continental System," the Napoleonic proposal for a Europe-wide embargo on British trade in manufactured goods. Napoleon's march on Moscow was slowed by Russian resistance at Borodino, where Marshal Mikhail Kutuzov's army was driven out of fortified redoubts after a day of destructive fighting. Although French forces were briefly able to seize control of Moscow, the subsequent retreat through the worst of the Russian winter ruined Napoleon's Grand Army.

Sino-Japanese War (1937-1945)

Japan's invasion of China was the primary cause of World War II in East Asia. As early as 1931, Japanese forces occupied Manchuria and set up a puppet state called Manchukuo. The Marco Polo Bridge Incident of 1937 resulted in open war between Japan and China. Japanese forces committed notorious atrocities during the invasion of China, including the 1937 massacres known as the Rape of Nanking. The Chinese war effort was hindered by internal conflict between Chiang Kai-Shek's Nationalist Kuomintang government and the communist insurgency led by Mao Zedong. Between 1942 and 1945 central China was largely cut off from the outside world due to Japanese conquest of coastal ports; Allied support was limited to air power deployed over the Himalayas from India, including the fighter pilots known as the "Flying Tigers."

Fall of Singapore (February 1942)

Japan's offensive in southeast Asia also struck at the British Empire. Japanese aircraft sank the British battleship Prince of Wales and drove obsolete British aircraft from the skies over the Malay Peninsula. Allied troops were driven back toward Singapore, Britain's major base in the Far East. Reinforcements from Britain and Australia arrived too late to repair the situation, and British general Arthur Percival was forced to surrender in February 1942. The loss of Singapore stunned the British Empire. Many Indian prisoners captured at Singapore switched sides to fight for the Japanese; British and Australian POWs labored in terrible conditions on the Siam-Burma railway depicted in the novel The Bridge over the River Kwai.

Attack on Pearl Harbor (December 1941)

Japanese plans for domination of the Pacific called for a surprise attack on the U.S. Navy base at Pearl Harbor in Hawaii; wrecking the American Pacific Fleet would allow the Japanese navy to invade southeast Asia with minimal opposition. Admiral Isoroku Yamamoto planned the Japanese strike, which sent carrier-based dive bombers and torpedo bombers to Oahu on Sunday, December 7, 1941. Four American battleships, including the USS Arizona, were sunk; the American aircraft carriers were at sea away from Pearl Harbor at the time of the attack. The losses sustained at Pearl Harbor shocked the previously neutral United States into entering World War II. President Franklin Roosevelt's address to Congress described December 7 as a "date which will live in infamy."

Alesia (52 BC)

Julius Caesar defeated the Celtic peoples of Gaul, establishing Roman rule of the lands beyond the Alps. The battle began when Caesar besieged Vercingetorix in the town of Alesia, shortly after the Roman defeat at Gergovia. The Romans built a wall to surround the city (a "circumvallation") and a second wall around that (a "contravallation") to protect themselves from the Gaulish relief army under Commius. When Commius launched a massive attack on the Romans, Caesar was able to defeat him and force the surrender of Vercingetorix. Although the Romans were outnumbered by as much as four to one, they proved victorious in what was the turning point of the Gallic Wars.

Las Navas de Tolosa (1212)

Las Navas de Tolosa was a major turning point in the Reconquista of the Iberian peninsula. A Christian alliance of Alfonso VIII of Castile, Sancho VII of Navarre, Afonso II of Portugal, and Peter II of Aragon defeated Muhammad al-Nasir and the Almohads.

Austerlitz (December 2, 1805, Czech Republic)

Napoleon's Grand Army then struck east at Austria and Russia, the land-bound members of the Third Coalition. Napoleon's first move was to force the surrender of 30,000 Austrians under General Mack at Ulm. The French then turned east into the heart of Austria, where they seized Vienna and awaited counterattack by the Russians. At Austerlitz on December 2, 1805, a mostly-Russian coalition army collided with the waiting French. (The Russians, led in person by Tsar Alexander I, were joined by the scattered remains of the Austrian army under Holy Roman Emperor Francis II. Austerlitz is thus known as the "Battle of the Three Emperors.") The allies, planning to advance their left, abandoned the Pratzen Heights, a dominating hill in the center of the battlefield. Napoleon seized the heights, splitting the Russian army and then defeating each half in turn. The resulting Peace of Pressburg (December 26, 1805) ended the War of the Third Coalition and brought about the downfall of the Holy Roman Empire.

Waterloo (June 18, 1815, Belgium)

Napoleon's escape from Elba began a period known as the "Hundred Days," in which the emperor briefly returned to the throne of France. The struggle between the restored emperor and the "Seventh Coalition" began when Napoleon's Army of the North marched into the Low Countries, hoping for a showdown with the British, Dutch, and Prussians before the Austrian and Russian armies gathering further east could come to their aid. The French brushed aside Allied advance guards at the two preliminary battles Quatre Bras and Ligny on June 16. Napoleon's victory over the Prussians at Ligny led him to falsely believe that he had enough time to pursue and defeat the British without further Prussian interference. On June 18 Napoleon's advance on Brussels approached the crossroads of Mont St. Jean, where the Duke of Wellington had set up a defensive position for a combined army of British Peninsular War veterans, Dutch, and pro-British Germans. On the French left, British troops defended the walled farm Hougoumont from a series of infantry assaults; in the center, Marshal Michel Ney's massed cavalry charge was broken by the square formations of the British infantry; on the right, Gebhard von Blücher's Prussian army arrived to attack the French army in the flank. Napoleon's final gamble was to commit his Imperial Guard to a renewed assault on the Allied center. The guardsmen were cut down by the fire of British light infantry, leading to the general collapse of the French army. Napoleon was exiled once more, this time to the isolated South Atlantic island St. Helena, where he died in 1821.

Marathon (490 BC)

Persian King Darius I's invasion of mainland Greece ended with a decisive victory for Miltiades and the Athenians at _____________. The defeated Persian commanders were Datis and Artaphernes. Among the few Athenian dead of the battle were archon Callimachus and the general Stesilaos. Legend has it that the Greek messenger Pheidippides ran to Athens with news of the victory, but collapsed upon arrival. This is the inspiration for the modern race known as the "marathon."

Pythagorean triples

Sets of small integers that satisfy the equation of the Pythagorean theorem, a^2 + b^2 = c^2, and could therefore be the side lengths of a right triangle. The simplest ones are {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, and {8, 15, 17}.

Battle of Adrianople (AD 378)

Taking place near what is now Edirne, Turkey, the Battle of ________________ signalled the beginning of the spread of Germanic peoples into the Western Roman Empire. The Romans were led by the eastern emperor Valens, while the Goths were led by Fritigern. Eager for glory, Valens decided not to wait on reinforcements from the western emperor Gratian, and instead attacked the Goths. In the battle, over two-thirds of the Roman army was killed, including Valens. The battle was chronicled by Ammianus Marcellinus, who thought it so important that he ended his history of the Roman Empire with the battle.

Battle of Leyte Gulf (October 1944)

The American landings on the island of _______ in the fall of 1944 fulfilled Douglas MacArthur's promise to return to the Philippines. Most of the Imperial Japanese Navy's remaining strength emerged to challenge the Allied landing, setting up one of the largest naval battles in world history. American admiral William "Bull" Halsey was criticized for the poorly-coordinated response to the Japanese attack. Nevertheless, American ships, aircraft, and submarines were able to destroy more than two dozen Japanese vessels, including the giant battleship Musashi. The Japanese position was by now so desperate that Japan began the practice of suicidal kamikaze attacks on Allied naval vessels.

Grunwald/Tannenberg (1410)

The Battle of Grunwald marked a major turning point in Eastern European history as it led to the decline of the Teutonic Knights and the rise of Poland-Lithuania. Grand Master Ulrich von Jungingen led the Teutonic Knights, and King Władysław II Jagiello led Polish forces and Grand Duke Vytautas led the Lithuanians. After their defeat, the Teutonic Knights held out in Marienburg Castle until the Peace of Thorn in 1411.

Battle of the Coral Sea (May 1942)

The Battle of the ________ _____ resulted from Japanese ambitions to invade Port Moresby, an Allied base in New Guinea. Frank Jack Fletcher's American fleet damaged two Japanese aircraft carriers, Shokaku and Zuikaku. The U.S. Navy's major loss was the carrier Lexington. The battle, a tactical draw, prevented the Japanese from attacking Port Moresby. It is notable for being the first naval battle fought entirely by aircraft — neither fleet was ever in visual range of the other's ships.

Manzikert (1071)

The Byzantine defeat at Manzikert (Malazgirt in Turkish) by the Seljuk Turks was a severe blow to the Byzantine empire's control of Anatolia. The Seljuks were led by Alp Arslan, while Emperor Romanus IV led the Byzantines. The Byzantines split their forces in half before the battle by sending general Joseph Tarchaneiotes to Khliat, and were further weakened by the desertion of Andronicus Dukas. The Byzantine emperor was captured in the battle, but released by Alp Arslan after signing a peace treaty. Romanus IV was later deposed and blinded.

Tours (732)

The Franks' victory at the Battle of Tours was a major turning point in the history of Europe. After conquering Iberia, the Umayyads advanced into France as far as Poitiers. The Franks were led by Charles Martel, while the Umayyads were led by Abdul Rahman Al Ghafiqi. The battle was preceded by a pact between the Berber governor Munuza and Odo, Duke of Aquitaine, though Odo was present at Tours. The best contemporary source for the battle is the Mozarabic Chronicle of 754.

Zama (202 BC)

The final battle of the Second Punic War, ________ was fought near Carthage in what is now Tunisia. Scipio Africanus's victory at the Battle of the Great Plains in 203 BC forced Hannibal to leave Italy and return to North Africa for the final showdown. Prior to the battle, the Numidian king Masinissa switched sides, and brought his considerable cavalry force to join the Romans. This, coupled with Scipio's strategy of opening up his lines to allow Carthaginian elephants through without harming his troops, led to a complete Roman victory.

Lake Peipus (1242)

The Russian victory at this battle ended territorial claims of the Teutonic Knights on Russian soil. The Russians were led by Alexander Nevsky and Andrey Yaroslavich, while the Livonian Order (a branch of the Teutonic Order) was led by Hermann of Dorpat. It is also known as the Battle on the Ice because it was fought on frozen Lake Peipus.

Similar figure

The areas are related by the square of any corresponding length, and the volumes are related by the cube of any corresponding length. For instance, if a square has a diagonal that is 30% longer than another square, it has an area that is 1.30 × 1.30 = 1.69 times as great (69% greater). Similar reasoning applies to perimeters, side lengths, diameters, and so forth.

Marengo (June 14, 1800, northern Italy)

The coup of 18 Brumaire (November 9, 1799) brought down France's existing government (the Directory) and made Napoleon himself "first consul," the effective leader of France. While Napoleon was absent in Egypt, a coalition of Austria, Russia, and Britain had pushed French troops back on all fronts. In 1800 Napoleon marched over the Alps to roll back Austrian gains in Italy. His troops, overextended in an attempt to relieve the Austrian siege of Genoa, were hit by an Austrian surprise attack on June 14, 1800. General Louis Desaix led a column of French reinforcements to Napoleon's aid; the additional troops drove off the Austrian army, but Desaix was shot and killed.

Leipzig (October 16-19, 1813, Germany)

The following year the Russian army marched west into central Europe at the head of a "Sixth Coalition" that brought the previously defeated Austrians and Prussians back into hostilities against France. At Leipzig in central Germany, coalition forces met a hastily-assembled replacement army raised by Napoleon after the disaster in Russia. More than 600,000 men fought in this four-day struggle, popularly known as the "Battle of the Nations" for the multi-ethnic nature of the coalition army. The forces of Saxony, one of the minor German states, switched sides during the Battle of Leipzig, leaving Napoleon's army to join the allies. The premature destruction of a bridge over the River Elster hindered Napoleon's retreat from Leipzig, the first in a series of military disasters that led to the emperor's forced abdication and exile to the Mediterranean island Elba in 1814.

Cannae (216 BC)

The largest battle of the Second Punic War, ___________ represented one of the worst defeats in Roman history. The Carthaginians were led by Hannibal, while the Romans were led by the consuls Lucius Aemilius Paullus and Gaius Terentius Varro. Hannibal employed a double-envelopment tactic, surrounded the Roman army, and destroyed it. Although a total disaster for the Romans, it resulted in their adopting of the Fabian strategy, in which battles are avoided in favor of a war of attrition. This eventually wore down Hannibal's army, and the Carthaginians had to leave Italy.

Salamis (480 BC)

The naval battle at ___________was a major turning point in the Greco-Persian Wars, as it signaled the beginning of the end of Persian attempts to conquer Greece. The battle is named after an island in the Saronic Gulf near Athens. Xerxes was so confident in victory that he watched the battle from a throne on the slopes of Mount Aegaleus. The Athenian general Themistocles devised a plan to lure the large, slow Persian ships into the narrow straits, where the Greek ships were able to outmaneuver and destroy much of the Persian fleet. The Persian admiral Ariabignes was killed in hand-to-hand combat, and the Queen of Halicarnassus, Artemisia, had to sink some of her allies' ships to escape.

Agincourt (1415)

The outnumbered army of King Henry V of England defeated the army of King Charles VI of France under Charles d'Albret. French commanders Jean le Maingre and Charles d'Orleans were captured. The battle occurred about one month after the English victory at the Siege of Harfleur. Five years after Agincourt, Henry was recognized as the heir to the French throne through the Treaty of Troyes.

Kosovo (1389)

This battle between Serbians and Ottomans eventually led to the subjugation of Serbia. Both Prince Lazar of Serbia and Sultan Murad I died during the battle. Bayezid I, who was present at the battle, became the Ottoman Sultan once his father died.

Guadalcanal Campaign (August 1942 to February 1943)

This first Allied counteroffensive in the Pacific targeted __________________, an island in the Solomons, to secure communications between the U.S. Pacific coast and Australia. "Operation Watchtower" was the codename for the initial U.S. Marine landings, which secured an airbase at Henderson Field and held off a Japanese counterattack on Edson's Ridge. The naval battle of Savo Island took place between American ships and Japanese forces (the so-called "Tokyo Express") trying to resupply troops on Guadalcanal. By early 1943, Allied dominance of the seas and skies around Guadalcanal forced Japan to withdraw its remaining troops.

Combinations

When we want to find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT, order doesn't matter; We say '3 choose 2' and write 3_C_2. 3_C_2 = 3_P_2/2!

Salamanca (July 22, 1812, Spain)

While Napoleon struggled against Austrians, Russians, and Prussians on the plains of central and eastern Europe, a smaller but no less violent conflict was fought for control of Spain and Portugal. During this "Peninsular War" (1807-1814), the throne of Spain was claimed by Napoleon's older brother, Joseph Bonaparte. Spanish and Portuguese resistance (the first "guerrilla" warfare, from the Spanish for "little war") was supported by the landing of British troops under the command of the Duke of Wellington. Wellington's triumph over Marshal Auguste Marmont at Salamanca in July 1812 was a decisive blow against the stability of Joseph's regime.

Battle of Iwo Jima (February to March 1945)

_____ ______ is an isolated volcanic island between the Marianas and Honshu; early in 1945 it was strategically important as a base for American air attacks on the Japanese home islands. The defenders were led by General Tadamichi Kuribayashi, who attempted to prolong the battle indefinitely by digging an elaborate system of tunnels to protect his troops. The U.S Marines who stormed the island starting in February 1945 sustained thousands of casualties. Joe Rosenthal's photograph of Marines raising the flag on Iwo's Mount Suribachi is one of the best known images of the Second World War.

Battle of Midway (June 1942)

___________ is considered the turning point of World War II in the Pacific. Japanese Admiral Yamamoto launched attacks on both Midway Island — an atoll northwest of Hawaii — and the Aleutian Islands in Alaska, but the Americans had broken the Japanese naval code and were forewarned. Admiral Chester W. Nimitz's American fleet lost the aircraft carrier Yorktown, but American dive-bombers destroyed four Japanese carriers: Soryu, Akagi, Kaga, and Hiryu. The loss of four irreplaceable aircraft carriers and the death of the best-trained Japanese pilots crippled Japanese naval aviation for the duration of the war.

Volume of a parallelepiped

abc

Actium (31 BC)

the fleet of Octavian defeated the combined forces of Cleopatra and Mark Antony at this battle near what is now Preveza in the Ambracian Gulf of Greece. Marcus Agrippa commanded Octavian's fleet, which consisted of small, nimble Liburnian ships. Antony's fleet consisted of massive Quinqueremes, which were less mobile. Following his victory in the battle, Octavian titled himself Princeps, and later Augustus. To some, Actium signals the end of the Roman Republic.

Treaty of Paris of 1898

ended the Spanish-American War and transferred Guam, the Philippines, and Puerto Rico to the U.S. while making Cuba (ostensibly) independent. The treaty was the beginning of American imperialism, and underwent a lengthy and contentious ratification.

Complex number

i = square root of -1

Battle of Aegospotami (405 BC)

on the Hellespont (Dardanelles) ended the Peloponnesian War and the Athenian Empire. After a setback at the Battle of Arginusae in 406 BC, the Spartans reinstated Lysander as the commander of their fleet. The result was a complete victory for Sparta; only a fraction of the Athenian fleet survived, including the general Conon, and the ship Paralus, which brought the news of defeat to Athens. Following the battle, the Spartans besieged Athens and forced its surrender.

Napoleon Bonaparte (born 1769 on the Mediterranean island Corsica)

rose to power during the later stages of the French Revolution. He defended the revolutionary government with artillery fire during the coup of 13 Vendémiaire (October 1795) and was rewarded with command of the French forces in Italy, where his series of battlefield victories forced Austria to sign the 1797 Treaty of Campo Formio. Over the next eighteen years, Napoleon rose from general to first consul to emperor of France, effectively ending the French Revolution and provoking a series of "Napoleonic Wars" that sent French troops into battle in places as distant as Egypt, Portugal, and central Russia. To defeat and depose the upstart emperor, it took a coalition of European powers nearly two decades of warfare, including the following battles:

Battle of Chalons (or Catalaunian Fields) (AD 451)

was an epic battle between the Romans and the Huns fought in what is now France. The Roman army was commanded by Flavius Aetius and included Visigoths under Theodoric I, who was killed by an Ostrogoth during the battle. The Hunnic army was led by Attila, who was rampaging through Gaul. The battle ended with a victory for the Roman-Visigothic alliance, which stopped the Huns' advance into Gaul. The next year, Attila invaded Italy; however, in 453, Attila died and his empire broke up shortly after.

Battle of Kadesh (1274 BC)

was fought near the Orontes River in what is now Syria between Egyptian Pharaoh Ramses II and the Hittite king Muwatalli II. Although Ramses proclaimed a great victory for himself, he was lucky to achieve a stalemate after being ambushed by Hittite chariots. Kadesh was probably the largest chariot battle in history, with over 5,000 chariots engaged. The Egyptian chariots were smaller and faster than those used by the Hittites, which gave the Egyptians an advantage.

Battle of the Milvian Bridge (AD 312)

was part of the civil war that ensued when Maxentius usurped the throne of the western half of the Roman Empire from Constantine. Prior to the battle, Constantine supposedly had a vision of God promising victory to his forces if he painted his shields with the Chi-Rho, a Christian symbol. Constantine was indeed victorious, and Maxentius drowned in the Tiber River during the battle. Eventually, Constantine was able to abolish the Tetrarchy, become the sole ruler of the Roman Empire, and end persecution of Christians.

Thermopylae (480 BC)

was the first battle of the second Persian invasion of Greece. Although the Persians under Xerxes I and his general Mardonius defeated the Spartans, King Leonidas and his Spartan troops put up a heroic defense of the pass at Thermopylae (the "hot gates"). The Greeks were betrayed by Ephialtes, who told the Persians about a path that led behind the Spartans. The battle was part of Themistocles' plan to halt the advance of the Persians. The other part of his plan was to block the Persian navy at Artemisium; a battle occurred there simultaneously.

Issus (333 BC)

was the second major battle between Alexander the Great and the Persian Empire, and the first to feature Darius III. The battle was fought along the Pinarus River near what is now Iskenderun in Turkey's Hatay province. Before the battle, Darius was able to surprise Alexander and cut him off from the main force of Macedonians. However, the battle ended with Darius fleeing the field and the capture of his tent and family. The battle was the subject of a 1528 painting by Albrecht Altdorfer, the leader of the Danube School.