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Battle of Tours

The Battle of Tours, also called the Battle of Poitiers and the Battle of the Highway of the Martyrs (معركة بلاط الشهداء), was fought on 10 October 732, and was an important battle during the Umayyad invasion of Gaul. It resulted in victory for the Frankish and Aquitanian forces, led by Charles Martel, over the invading Umayyad forces, led by Abd al-Rahman al-Ghafiqi, governor of al-Andalus. Several historians, such as Edward Gibbon, have credited the Christian victory in the battle as an important factor in curtailing the spread of Islam in Western Europe.

Details of the battle, including the number of combatants and its exact location, are unclear from the surviving sources. Most sources agree that the Umayyads had a larger force and suffered heavier casualties. Notably, the Frankish troops apparently fought without heavy cavalry. The battlefield was located somewhere between the cities of Poitiers and Tours, in northern Aquitaine in western France, near the border of the Frankish realm and the then-independent Duchy of Aquitaine under Odo the Great.

Al-Ghafiqi was killed in combat, and the Umayyad army withdrew after the battle. The battle helped lay the foundations of the Carolingian Empire and Frankish domination of western Europe for the next century. Most historians agree that "the establishment of Frankish power in western Europe shaped that continent's destiny and the Battle of Tours confirmed that power."

Background

The Battle of Tours followed two decades of Umayyad conquests in Europe which had begun with the invasion of the Christian Visigothic Kingdom of the Iberian Peninsula in 711. These were followed by military expeditions into the Frankish territories of Gaul, former provinces of the Roman Empire. Umayyad military campaigns reached northward into Aquitaine and Burgundy, including a major engagement at Bordeaux and a raid on Autun. Charles's victory is widely believed to have stopped the northward advance of Umayyad forces from the Iberian Peninsula and to have prevented the Islamization of Western Europe.

Most historians assume that the two armies met where the rivers Clain and Vienne join between Tours and Poitiers. The number of troops in each army is not known. The Mozarabic Chronicle of 754, a Latin contemporary source which describes the battle in greater detail than any other Latin or Arabic source, states that "the people of Austrasia [the Frankish forces], greater in number of soldiers and formidably armed, killed the king, Abd ar-Rahman", which agrees with many Arab and Muslim historians. However, virtually all Western sources disagree, estimating the Franks as numbering 30,000, less than half the Muslim force.

Some modern historians, using estimates of what the land was able to support and what Martel could have raised from his realm and supported during the campaign, believe the total Muslim force outnumbered the Franks if one counts the outlying raiding parties which rejoined the main body before Tours. Drawing on non-contemporary Muslim sources, Creasy describes the Umayyad forces as 80,000 strong or more. Writing in 1999, Paul K. Davis estimates the Umayyad forces at 80,000 and the Franks at about 30,000, while noting that modern historians have estimated the strength of the Umayyad army at Tours at between 20,000–80,000. However, Edward J. Schoenfeld, rejecting the older figures of 60,000–400,000 Umayyads and 75,000 Franks, contends that "estimates that the Umayyads had over fifty thousand troops (and the Franks even more) are logistically impossible." Similarly, historian Victor Davis Hanson believes both armies were roughly the same size, between 20,000 and 30,000 men.

Modern historical analysis may be more accurate than the medieval sources, as the modern figures are based on estimates of the logistical ability of the countryside to support these numbers of men and animals. Davis and Hanson point out that both armies had to live off the countryside, neither having a logistical system sufficient to provide supplies for a campaign. Other sources give the following estimates: "Gore places the Frankish army at 15,000–20,000, although other estimates range from 30,000 to 80,000. In spite of wildly varying estimates of the Muslim force, he places that army as around 20,000–25,000. Other estimates also range up to 80,000, with 50,000 not an uncommon estimate."

Losses during the battle are unknown, but chroniclers later claimed that Charles Martel's force lost about 1,500 while the Umayyad force was said to have suffered massive casualties of up to 375,000 men. However, these same casualty figures were recorded in the Liber Pontificalis for Duke Odo the Great's victory at the Battle of Toulouse (721). Paul the Deacon reported correctly in his History of the Lombards (written around 785) that the Liber Pontificalis mentioned these casualty figures in relation to Odo's victory at Toulouse (though he claimed that Charles Martel fought in the battle alongside Odo), but later writers, probably "influenced by the Continuations of Fredegar, attributed the Muslims casualties solely to Charles Martel, and the battle in which they fell became unequivocally that of [Tours-Poitiers]." The Vita Pardulfi, written in the middle of the eighth century, reports that after the battle 'Abd-al-Raḥmân's forces burned and looted their way through the Limousin region on their way back to Al-Andalus, which implies that they were not destroyed to the extent imagined in the Continuations of Fredegar.

Umayyads
The invasion of Hispania, and then Gaul, was conducted by the Umayyad dynasty (بنو أمية banū umayya / الأمويون al-umawiyyūn also "Umawi"), the first dynasty of Sunni caliphs of the Sunni Islamic empire after the reign of the Rashidun Caliphs (Abu Bakr, Umar, Uthman, and Ali) ended. The Umayyad Caliphate, at the time of the Battle of Tours, was perhaps the world's foremost military power. The great expansion of the Caliphate occurred under the reign of the Umayyads. Muslim armies pushed east across Persia and west across North Africa through the late 7th century.

The Umayyad empire was now a vast domain that ruled a diverse array of peoples. It had defeated and completely absorbed the Sasanian Empire, while also conquering much of the Byzantine Empire, including Syria, Armenia, and North Africa, although Leo the Isaurian stemmed the tide when his army defeated the Umayyads at the Battle of Akroinon (740), their final campaign in Anatolia.

Franks
The Frankish realm under Charles Martel was the foremost military power of western Europe. During most of his tenure in office as commander-in-chief of the Franks, the Frankish kingdom consisted of north and eastern France (Austrasia, Neustria and Burgundy), most of western Germany, and the Low Countries (Luxembourg, Belgium and the Netherlands). This domain had begun to progress towards becoming the first real imperial power in western Europe since the fall of Rome. However, it continued to struggle against external forces such as the Saxons, Frisians, and other opponents such as the Basque-Aquitanians led by Odo the Great (Old French: Eudes, or Eudo), Duke of Aquitaine and Vasconia.

Umayyad conquests from Hispania

The Umayyad troops, under Al-Samh ibn Malik al-Khawlani, the governor-general of al-Andalus, overran Septimania by 719, following their sweep up the Iberian Peninsula. Al-Samh set up his capital from 720 at Narbonne, which the Moors called Arbūna. With the port of Narbonne secure, the Umayyads swiftly subdued without much resistance the cities of Alet, Béziers, Agde, Lodève, Maguelonne, and Nîmes, still controlled by their Visigothic counts.

The Umayyad campaign into Aquitaine suffered a temporary setback at the Battle of Toulouse. Duke Odo the Great broke the siege of Toulouse, taking Al-Samh ibn Malik's forces by surprise. Al-Samh ibn Malik was mortally wounded. This defeat did not stop incursions into old Roman Gaul, as Moorish forces, soundly based in Narbonne and easily resupplied by sea, struck eastwards in the 720s, penetrating as far as Autun in Burgundy in 725.

Threatened by both the Umayyads in the south and by the Franks in the north, in 730 Odo allied himself with the Berber commander Uthman ibn Naissa, called "Munuza" by the Franks, the deputy governor of what would later become Catalonia. To seal the alliance, Uthman was given Odo's daughter Lampagie in marriage, and the Moors ceased their raids across the Pyrenees, Odo's southern border. However, the next year, the Berber leader killed the bishop of Urgell Nambaudus and detached himself from his Arab masters in Cordova. Abd Al Raḥman in turn sent an expedition to crush his revolt, and next directed his attention against Uthman's ally Odo.

Odo collected his army at Bordeaux, but was defeated, and Bordeaux was plundered. During the following Battle of the River Garonne, the Chronicle of 754 commented that "God alone knows the number of the slain". The chronicle added that they "pierced through the mountains, trampled over rough and level ground, plundered far into the country of the Franks, and smote all with the sword, insomuch that when Eudo came to battle with them at the River Garonne, he fled."

Odo's appeal to the Franks
Odo set about reorganizing his troops despite his heavy losses, and gave the Frankish leader notice of the impending danger to the heartland of his realm while appealing to the Franks for assistance, which Charles Martel only granted after Odo agreed to submit to Frankish authority.

It appears that the Umayyads were not aware of the true strength of the Franks. The Umayyad forces were not particularly concerned about any of the Germanic tribes, including the Franks, and the Arab chronicles of that age show that awareness of the Franks as a growing military power only came after the Battle of Tours.

Further, the Umayyads appear not to have scouted northward for potential foes, for if they had, they surely would have noted Charles Martel as a force to be reckoned with on his own, because of his growing domination of much of Europe since 717.

Umayyad advance towards the Loire
In 732, the Umayyad advance force was proceeding north towards the Loire River, having outpaced their supply wagons and a large part of their army. Having easily destroyed all resistance in that part of Gaul, the invading army had split off into several raiding parties, while the main body advanced more slowly.

The Umayyads delayed their campaign late in the year probably because the army needed to live off the land as they advanced. They had to wait until the area's wheat harvest was ready and then until a reasonable amount of the harvest had been stored.

Odo was defeated so easily at Bordeaux and Garonne, despite winning eleven years earlier at the Battle of Toulouse, because at Toulouse he had managed a surprise attack against an overconfident and unprepared foe: the Umayyad forces were mostly infantry, and what cavalry they did have were never mobilized. As Herman of Carinthia wrote in one of his translations of a history of al-Andalus, Odo managed a highly successful encircling envelopment which took the attackers completely by surprise, resulting in the slaughter of the Muslim forces.

At Bordeaux and again at Garonne, the Umayyad forces were mostly cavalry and had the chance to mobilize, which led to the devastation of Odo's army. Odo's forces, like other European troops of that era, had no stirrups at that time and therefore no heavy cavalry. Most of their troops were infantry. The Umayyad heavy cavalry broke Odo's infantry in their first charge and then slaughtered them as they fled.

The invading force went on to devastate southern Gaul. A possible motive, according to the second continuator of the Chronicle of Fredegar, were the riches of the Abbey of Saint Martin of Tours, the most prestigious and holiest shrine in western Europe at the time. Upon hearing this, Austrasia's Mayor of the Palace, Charles Martel, prepared his army and marched south, avoiding the old Roman roads, hoping to take the Muslims by surprise.

Preparations and manoeuver
By all accounts, the invading forces were caught off guard to discover a large force sitting directly in their path to Tours. Charles achieved the total surprise he had hoped for. He then chose not to attack but rather set up a defensive, phalanx-like formation. According to Arab sources, the Franks drew up in a large square, with hills and trees in their front to impede or break up Muslim cavalry charges.

For one week, the two armies engaged in minor skirmishes while the Umayyads waited for their full strength to arrive. 'Abd-al-Raḥmân, despite being a proven commander, had been outmaneuvered by allowing Charles to concentrate his forces and pick the field of battle. Furthermore, the Umayyads could not judge the size of Charles' army since he had used the forest to conceal his true numbers.

Charles' infantry was his best hope for victory. Seasoned and battle-hardened, most of them had fought under him for years, some as far back as 717. In addition to his army, he also had levies of militia which had not seen significant military use except for gathering food and harassing the Muslim army.

While many historians through the centuries have believed that the Franks were outnumbered at the onset of battle by at least two to one, some sources, such as the Mozarabic Chronicle of 754, contradict that assertion.

Charles correctly assumed that 'Abd-al-Raḥmân would feel compelled to give battle, and move on and try to loot Tours. Neither side wanted to attack. Abd-al-Raḥmân felt he had to sack Tours, which meant he had to go through the Frankish army on the hill in front of him. Charles' decision to stay in the hills proved crucial, as it forced the Umayyad cavalry to charge uphill and through trees, diminishing their effectiveness.

Charles had been preparing for this confrontation since the Battle of Toulouse a decade earlier. Gibbon believes, as do most historians, that Charles had made the best of a bad situation. Though allegedly outnumbered and without any heavy cavalry, he had tough, battle-hardened infantrymen who believed in him implicitly. At a time in the Middle Ages when permanent armies had been non-existent in Europe, Charles even took out a large loan from the Pope after convincing him of the impending emergency, to properly train and maintain a full-size army largely composed of professional infantry. Moreover, as Davis points out, these infantrymen were heavily armed.

Formed into a phalanx formation, they were able to withstand a cavalry charge better than might be expected, especially as Charles had secured the high ground – with trees before him to further impede any cavalry charges. The failure of Arab intelligence extended to the fact that they were unaware of how good his forces were; he had trained them for a decade. And while he was well aware of the Caliphate's strengths and weaknesses, he was aware they knew nothing about the Franks.

Furthermore, the Franks were dressed for the cold, whereas the Arabs wore very light clothing more suitable for North African winters than European winters.

The battle eventually became a waiting game in which the Muslims did not want to attack an army that could be numerically superior and wanted the Franks to come out into the open. The Franks formed up in a thick defensive formation and waited for them to charge uphill. The battle finally began on the seventh day, as 'Abd-al-Raḥmân did not want to wait any longer, with winter approaching.

Engagement

'Abd-al-Raḥmân trusted in the tactical superiority of his cavalry and had them charge repeatedly throughout the day. The disciplined Frankish soldiers withstood the assaults, though according to Arab sources, the Arab cavalry broke into the Frankish square several times. Despite this, the Franks did not break. The well-trained Frankish soldiers accomplished what was not thought possible at that time: infantry withstanding a heavy cavalry charge. Paul Davis says the core of Charles' army was a professional infantry that was both highly disciplined and well-motivated, "having campaigned with him all over Europe".

Contemporary accounts
The Mozarabic Chronicle of 754 "describes the battle in greater detail than any other Latin or Arabic source". It says of the encounter that, "While Abd ar-Rahman was pursuing Odo, he decided to despoil Tours by destroying its palaces and burning its churches. There he confronted the consul of Austrasia by the name of Charles, a man who, having proved himself to be a warrior from his youth and an expert in things military, had been summoned by Odo. After each side had tormented the other with raids for almost seven days, they finally prepared their battle lines and fought fiercely. The northern peoples remained as immobile as a wall, holding together like a glacier in the cold regions. In the blink of an eye, they annihilated the Arabs with the sword. The people of Austrasia, greater in a number of soldiers and formidably armed, killed the king, Abd ar-Rahman, when they found him, striking him on the chest. But suddenly, within sight of the countless tents of the Arabs, the Franks despicably sheathed their swords postponing the fight until the next day since night had fallen during the battle. Rising from their own camp at dawn, the Europeans saw the tents and canopies of the Arabs all arranged just as they had appeared the day before. Not knowing that they were empty and thinking that inside them there were Saracen forces ready for battle, they sent officers to reconnoiter and discovered that all the Ishmaelite troops had left. They had indeed fled silently by night in tight formation, returning to their own country."

Charles Martel's family composed, for the fourth book of the Continuations of Fredegar's Chronicle, a stylized summary of the battle:

"Prince Charles boldly drew up his battle lines against them [the Arabs] and the warrior rushed in against them. With Christ's help, he overturned their tents and hastened to battle to grind them small in slaughter. The king Abdirama having been killed, he destroyed [them], driving forth the army, he fought and won. Thus did the victor triumph over his enemies."

This source details further that "he (Charles Martel) came down upon them like a great man of battle". It goes on to say Charles "scattered them like the stubble".

The Latin word used for "warrior", belligerator, "is from the Book of Maccabees, chapters 15 and 16", which describe huge battles.

It is thought that Bede's Ecclesiastical History of the English People (Book V, Chapter XXIV) includes a reference to the Battle of Tours: "...a dreadful plague of Saracens ravaged France with miserable slaughter, but they not long after in that country received the punishment due to their wickedness".

Strategic analysis
Gibbon makes the point that 'Abd-al-Raḥmân did not move at once against Charles Martel, and was surprised by him at Tours as Charles had marched over the mountains while avoiding the roads to surprise the Muslim invaders. Thus, Charles selected the time and place they would fight.

'Abd-al-Raḥmân was a good general, but failed to do two things he should have done before the battle:
 * He either assumed that the Franks would not come to the aid of their Aquitanian rivals, or did not care, and he thus failed to assess their strength before the invasion.
 * He failed to scout the movements of the Frankish army.

These failures disadvantaged the Muslim army in the following ways:
 * The invaders were burdened with booty that played a role in the battle.
 * They had casualties before they fought the battle.
 * Weaker opponents such as Odo were not bypassed, whom they could have picked off at will later, while moving at once to force battle with the real power in Europe and at least partially pick the battlefield.

While some military historians point out that leaving enemies in your rear is not generally wise, the Mongols proved that indirect attack, and bypassing weaker foes to eliminate the strongest first, can be a devastatingly effective mode of invasion. In this case, those enemies were virtually no danger, given the ease with which the Muslims destroyed them. The real danger was Charles, and the failure to scout Gaul adequately was disastrous.

According to Creasy, both western and Muslim histories agree the battle was hard fought, and that the Umayyad heavy cavalry had broken into the square, but agreed that the Franks were in formation still strongly resisting.

Charles could not afford to stand idly by while Frankish territories were threatened. He would have to face the Umayyad armies sooner or later, and his men were enraged by the utter devastation of the Aquitanians and wanted to fight. But Sir Edward Creasy noted that,

"When we remember that Charles had no standing army, and the independent spirit of the Frank warriors who followed his standard, it seems most probable that it was not in his power to adopt the cautious policy of watching the invaders, and wearing out their strength by delay. So dreadful and so widespread were the ravages of the Saracenic light cavalry throughout Gaul, that it must have been impossible to restrain for any length of time the indignant ardor of the Franks. And, even if Charles could have persuaded his men to look tamely on while the Arabs stormed more towns and desolated more districts, he could not have kept an army together when the usual period of a military expedition had expired."

Both Hallam and Watson argue that had Charles failed, there was no remaining force to protect Western Europe. Hallam perhaps said it best: "It may justly be reckoned among those few battles of which a contrary event would have essentially varied the drama of the world in all its subsequent scenes: with Marathon, Arbela, the Metaurus, Châlons and Leipzig."

Strategically, and tactically, Charles probably made the best decision he could in waiting until his enemies least expected him to intervene, and then marching by stealth to catch them by surprise at a battlefield of his choosing. Probably he and his own men did not realize the seriousness of the battle they had fought, as one historian put it: "few battles are remembered over 1,000 years after they are fought, but the Battle of [Tours-Poitiers] is an exception ... Charles Martel turned back a Muslim raid that had it been allowed to continue, might have conquered Gaul." Roger Collins disputes interpretations of ever-expanding Umayyad forces, reminding their internal cohesion problems and the capture of Autun in 725, when the Burgundian stronghold was captured and sacked, then just abandoned by Anbasa's raiding forces.

Umayyad retreat and second invasion
The Umayyad army retreated south over the Pyrenees. Charles continued to expand south in subsequent years. After the death of Odo (c. 735), who had reluctantly acknowledged Charles' suzerainty in 719, Charles wished to unite Odo's duchy to himself and went there to elicit the proper homage of the Aquitanians. But the nobility proclaimed Hunald, Odo's son, as the duke, and Charles recognized his legitimacy when the Umayyads entered Provence as part of an alliance with Duke Maurontus the next year.

Hunald, who originally resisted acknowledging Charles as an overlord, soon had little choice. He acknowledged Charles as his overlord, albeit not for long, and Charles confirmed his duchy.

Umayyad invasion (735–39)
In 735, Uqba ibn al-Hajjaj, the new governor of al-Andalus, invaded Gaul. Antonio Santosuosso and other historians detail how he advanced into France to avenge the defeat at Tours and to spread Islam. According to Santosuosso, Uqba ibn al-Hajjaj converted about 2,000 Christians he had captured over his career. In the last major attempt at an invasion of Gaul through Iberia, a sizable expedition was assembled at Saragossa and entered what is now French territory in 735, crossed the River Rhone, and captured and looted Arles. From there, he struck into the heart of Provence, ending with the capture of Avignon, despite strong resistance.

Uqba ibn al-Hajjaj's forces remained in Septimania and part of Provence for four years, carrying raids to Lyons, Burgundy, and Piedmont. Charles Martel invaded Septimania in two campaigns in 736 and 739, but was forced back again to Frankish territory under his control. Alessandro Santosuosso strongly argues that the second (Umayyad) expedition was probably more dangerous than the first. The second expedition's failure put an end to any serious Muslim expedition across the Pyrenees, although raids continued. Plans for further large-scale attempts were hindered by internal turmoil in the Umayyad lands which often made enemies out of their own kind.

Advance to Narbonne
Despite the defeat at Tours, the Umayyads remained in control of parts of Septimania for another 27 years, though they could not expand further. The treaties reached earlier with the local population stood firm and were further consolidated in 734 when the governor of Narbonne, Yusuf ibn Abd al-Rahman al-Fihri, concluded agreements with several towns on common defense arrangements against the encroachments of Charles Martel, who had systematically brought the south to heel as he extended his domains. He conquered Umayyad fortresses and destroyed their garrisons at the Siege of Avignon and the Siege of Nîmes.

The army attempting to relieve Narbonne met Charles in open battle at the Battle of the River Berre and was destroyed. However, Charles failed in his attempt to take Narbonne at the Siege of Narbonne in 737, when the city was jointly defended by its Muslim Arab and Berber, and its Christian Visigothic citizens.

Carolingian dynasty
Reluctant to tie down his army for a siege that could last years, and believing he could not afford the losses of an all-out frontal assault such as he had used at Arles, Charles was content to isolate the few remaining invaders in Narbonne and Septimania. The threat of invasion was diminished after the Umayyad defeat at Narbonne, and the unified Caliphate would collapse into civil war in 750 at the Battle of the Zab.

It was left to Charles' son, Pepin the Short, to force Narbonne's surrender in 759, thus bringing it into the Frankish domains. The Umayyad dynasty was expelled, driven back to Al-Andalus where Abd al-Rahman I established an emirate in Córdoba in opposition to the Abbasid caliph in Baghdad.

In the northeast of Spain, the Frankish emperors established the Marca Hispanica across the Pyrenees in part of what today is Catalonia, reconquering Girona in 785 and Barcelona in 801. This formed a buffer zone against Muslim lands across the Pyrenees. Historian J.M. Roberts said in 1993 of the Carolingian dynasty:

"It produced Charles Martel, the soldier who turned the Arabs back at Tours, and the supporter of Saint Boniface the Evangelizer of Germany. This is a considerable double mark to have left on the history of Europe."

Before the Battle of Tours, stirrups may have been unknown in the west. Lynn Townsend White Jr. argues that the adoption of the stirrup for cavalry was the direct cause of the development of feudalism in the Frankish realm by Charles Martel and his heirs.

Historical and macrohistorical views
The historical views of this battle fall into three great phases, both in the East and especially in the West. Western historians, beginning with the Mozarabic Chronicle of 754, stressed the macrohistorical impact of the battle, as did the Continuations of Fredegar. This became a claim that Charles had saved Christianity, as Gibbon and his generation of historians agreed that the Battle of Tours was unquestionably decisive in world history.

Modern historians have essentially fallen into two camps on the issue. The first camp essentially agrees with Gibbon, and the other argues that the battle has been massively overstated – turned from a raid in force to an invasion, and from a mere annoyance to the Caliph to a shattering defeat that helped end the Islamic Expansion Era. It is essential, however, to note that within the first group, those who agree the battle was of macrohistorical importance, there are a number of historians who take a more moderate and nuanced view of the significance of the battle, in contrast to the more dramatic and rhetorical approach of Gibbon. The best example of this school is William E. Watson, who does believe the battle has such importance, as will be discussed below, but analyzes it militarily, culturally, and politically, rather than seeing it as a classic "Muslim versus Christian" confrontation.

In the East, Arab histories followed a similar path. First, the battle was regarded as a disastrous defeat; then, it largely faded from Arab histories, leading to a modern dispute which regards it as either a second loss to the great defeat of the Second Siege of Constantinople, where the Bulgarian Emperor Tervel played a crucial role, or a part of a series of great macrohistorical defeats which together brought about the fall of the first Caliphate. With the Byzantines and Bulgarians together with the Franks both successfully blocking further expansion, internal social troubles came to a head, starting with the Great Berber Revolt of 740, and ending with the Battle of the Zab, and the destruction of the Umayyad Caliphate.

In Western history
The first wave of modern historians, especially scholars on Rome and the medieval period, such as Edward Gibbon, contended that had Charles fallen, the Umayyad Caliphate would have easily conquered a divided Europe. Gibbon famously observed:

"A victorious line of march had been prolonged above a thousand miles from the rock of Gibraltar to the banks of the Loire; the repetition of an equal space would have carried the Saracens to the confines of Poland and the Highlands of Scotland; the Rhine is not more impassable than the Nile or Euphrates, and the Arabian fleet might have sailed without a naval combat into the mouth of the Thames. Perhaps the interpretation of the Koran would now be taught in the schools of Oxford, and her pulpits might demonstrate to a circumcised people the sanctity and truth of the revelation of Mahomet."

Nor was Gibbon alone in lavishing praise on Charles as the savior of Christendom and western civilization. H. G. Wells wrote: "The Moslim [sic] when they crossed the Pyrenees in 720 found this Frankish kingdom under the practical rule of Charles Martel, the Mayor of the Palace of a degenerate descendant of Clovis, and experienced the decisive defeat of [Tours-Poitiers] (732) at his hands. This Charles Martel was practically overlord of Europe north of the Alps from the Pyrenees to Hungary. He ruled over a multitude of subordinate lords speaking French-Latin and High and Low German languages."

Gibbon was echoed a century later by the Belgian historian Godefroid Kurth, who wrote that the Battle of Tours "must ever remain one of the great events in the history of the world, as upon its issue depended whether Christian Civilization should continue or Islam prevail throughout Europe."

German historians were especially ardent in their praise of Charles Martel; Schlegel speaks of this "mighty victory", and tells how "the arm of Charles Martel saved and delivered the Christian nations of the West from the deadly grasp of all-destroying Islam." Creasy quotes Leopold von Ranke's opinion that this period was

"one of the most important epochs in the history of the world, the commencement of the eighth century, when on the one side Mohammedanism threatened to overspread Italy and Gaul, and on the other the ancient idolatry of Saxony and Friesland once more forced its way across the Rhine. In this peril of Christian institutions, a youthful prince of Germanic race, Karl Martell, arose as their champion, maintained them with all the energy which the necessity for self-defense calls forth, and finally extended them into new regions."

The German military historian Hans Delbrück said of this battle "there was no more important battle in the history of the world." (The Barbarian Invasions, p. 441.) Had Charles Martel failed, Henry Hallam argued, there would have been no Charlemagne, no Holy Roman Empire or Papal States; all these depended upon Charles's containment of Islam from expanding into Europe while the Caliphate was unified and able to mount such a conquest. Another great mid era historian, Thomas Arnold, ranked the victory of Charles Martel even higher than the victory of Arminius in its impact on all of modern history: "Charles Martel's victory at Tours was among those signal deliverances which have affected for centuries the happiness of mankind." Louis Gustave and Charles Strauss said "The victory gained was decisive and final, The torrent of Arab conquest was rolled back and Europe was rescued from the threatened yoke of the Saracens."

Charles Oman concluded that:

"At [Tours-Poitiers] the Franks fought as they had done two hundred years before at Casilinum, in one solid mass, without breaking rank or attempting to maneuver. Their victory was won by the purely defensive tactics of the infantry square; the fanatical Arabs, dashing against them time after time, were shattered to pieces, and at last, fled under the shelter of night. But there was no pursuit, for Charles had determined not to allow his men to stir a step from the line to chase the broken foe."

John Bagnell Bury, writing at the beginning of the 20th century, said "The Battle of Tours ... has often been represented as an event of the first magnitude for the world's history, because after this, the penetration of Islam into Europe was finally brought to a standstill."

Modern Western historians are clearly divided on the importance of the battle, and where it should rank in military history; see below.

Adolf Hitler on the Battle of Tours
Albert Speer, Hitler's Armaments Minister, described how Hitler expressed approval of Islam, saying that Hitler had been particularly impressed by what he had heard from a delegation of Arabs. When the Arabs had tried to penetrate Central Europe in the 8th century, they had been driven back at the Battle of Tours; if they had won that battle, the world would have become Muslim (maybe). Hitler considered that Islam was more suited to the "Germanic" temperament and would have been more compatible to the Germans than Christianity.

In Muslim history
Eastern historians, like their Western counterparts, have not always agreed on the importance of the battle. According to Bernard Lewis, "The Arab historians, if they mention this engagement [the Battle of Tours] at all, present it as a minor skirmish," and Gustave von Grunebaum writes: "This setback may have been important from the European point of view, but for Muslims at the time, who saw no master plan imperiled thereby, it had no further significance." Contemporary Arab and Muslim historians and chroniclers were much more interested in the second Umayyad siege of Constantinople in 718, which ended in a disastrous defeat.

However, Creasy has claimed: "The enduring importance of the battle of Tours in the eyes of the Muslims is attested not only by the expressions of 'the deadly battle' and 'the disgraceful overthrow' which their writers constantly employ when referring to it but also by the fact that no more serious attempts at conquest beyond the Pyrenees were made by the Saracens."

Thirteenth-century Moroccan author Ibn Idhari al-Marrakushi, mentioned the battle in his history of the Maghrib, "al-Bayan al-Mughrib fi Akhbar al-Maghrib." According to Ibn Idhari, "Abd ar-Rahman and many of his men found martyrdom on the balat ash-Shuhada'i (the path of the martyrs)." Antonio Santosuosso points that "they (the Muslims) called the battle's location, the road between Poitiers and Tours, 'the pavement of Martyrs'." However, as Henry Coppée pointed out, "The same name was given to the battle of Toulouse and is applied to many other fields on which the Moslemah were defeated: they were always martyrs for the faith."

Khalid Yahya Blankinship argued that the military defeat at Tours was one of the failures that contributed to the decline of the Umayyad caliphate:

"Stretching from Morocco to China, the Umayyad caliphate based its expansion and success on the doctrine of jihad – armed struggle to claim the whole earth for God's rule, a struggle that had brought much material success for a century but suddenly ground to a halt followed by the collapse of the ruling Umayyad dynasty in 750 AD. The End of the Jihad State demonstrates for the first time that the cause of this collapse came not just from internal conflict, as has been claimed, but from a number of external and concurrent factors that exceeded the caliphate's capacity to respond. These external factors began with crushing military defeats at Byzantium, Toulouse, and Tours, which led to the Berber Revolt of 740 in Iberia and Northern Africa."

Supporting the significance of Tours as a world-altering event
Ninth-century chroniclers recorded the outcome of the battle as a divine judgment in favor of Charles and gave him the nickname Martellus ("The Hammer"). Later Christian chroniclers and pre-20th century historians praised Charles Martel as the champion of Christianity, characterizing the battle as the decisive turning point in the struggle against Islam, a struggle which preserved Christianity as the religion of Europe. According to modern military historian, Victor Davis Hanson "most of the 18th and 19th century historians like Gibbon saw Tours as a landmark battle that marked the high tide of the Muslim advance into Europe." Leopold von Ranke felt that Tours-Poitiers "was the turning point of one of the most important epochs in the history of the world."

William E. Watson writes that "the subsequent history of the West would have proceeded along vastly different currents had 'Abd ar-Rahman been victorious at Tours-Poitiers in 732" and that "[a]fter examining the motives for the Muslim drive north of the Pyrenees, one can attach a macrohistorical significance to the encounter ... especially when one considers the attention paid to the Franks in Arabic literature and the successful expansion of Muslims elsewhere in the medieval period."

Victorian writer John Henry Haaren says in Famous Men of the Middle Ages "The battle of Tours or Poitiers as it should be called is regarded as one of the decisive battles of the world. It decided that Christians and not Muslims should be the ruling power in Europe." Bernard Grun delivers this assessment in his "Timetables of History", reissued in 2004: "In 732 Charles Martel's victory over the Arabs at the Battle of Tours stems the tide of their westward advance."

Historian and humanist Michael Grant lists the battle of Tours in the macrohistorical dates of the Roman era. Historian Norman Cantor who specialized in the medieval period, teaching and writing at Columbia and New York University said in 1993: "It may be true that the Arabs had now fully extended their resources and they would not have conquered France, but their defeat (at Tours) in 732 put a stop to their advance to the North."

Military historian Robert W. Martin considers Tours "one of the most decisive battles in all of history." Additionally, historian Hugh Kennedy says "it was clearly significant in establishing the power of Charles Martel and the Carolingians in France, but it also had profound consequences in Muslim Spain. It signaled the end of the ghanima (booty) economy."

Military Historian Paul Davis argued in 1999 "had the Muslims been victorious at Tours, it is difficult to suppose what population in Europe could have organized to resist them." Likewise, George Bruce in his update of Harbottle's classic military history Dictionary of Battles maintains that "Charles Martel defeated the Moslem army effectively ending Moslem attempts to conquer western Europe."

History professor Antonio Santosuosso comments on Charles, Tours, and the subsequent campaigns against Rahman's son in 736–737, that these later defeats of invading Muslim armies were at least as important as Tours in their defense of Western Christendom and its monasteries, the centers of learning that ultimately led Europe out of her Middle Ages. He also makes an argument, after studying the Arab histories of the period, that these were armies of invasion sent by the Caliph not just to avenge Tours, but to begin the end of Christian Europe and bring it into the Caliphate.

Professor of religion Huston Smith says in The World's Religions: Our Great Wisdom Traditions "But for their defeat by Charles Martel in the Battle of Tours in 732, the entire Western world might today be Muslim." Historian Robert Payne on page 142 in The History of Islam said "The more powerful Muslims and the spread of Islam were knocking on Europe's door. And the spread of Islam was stopped along the road between the towns of Tours and Poitiers, France, with just its head in Europe."

Victor Davis Hanson has commented that

"Recent scholars have suggested [Tours-Poitiers], so poorly recorded in contemporary sources, was a mere raid and thus a construct of western mythmaking or that a Muslim victory might have been preferable to continued Frankish dominance. What is clear is that [Tours-Poitiers] marked a general continuance of the successful defense of Europe, (from the Muslims). Flush from the victory at Tours, Charles Martel went on to clear southern France from Islamic attackers for decades, unify the warring kingdoms into the foundations of the Carolingian Empire, and ensure ready and reliable troops from local estates."

Paul Davis, another modern historian, says "whether Charles Martel saved Europe for Christianity is a matter of some debate. What is sure, however, is that his victory ensured that the Franks would dominate Gaul for more than a century." Davis writes, "Moslem defeat ended the Moslems' threat to western Europe, and Frankish victory established the Franks as the dominant population in western Europe, establishing the dynasty that led to Charlemagne."

Objecting to the significance of Tours as a world-altering event
Other historians disagree with this assessment. Alessandro Barbero writes, "Today, historians tend to play down the significance of the battle of [Tours-Poitiers], pointing out that the purpose of the Muslim force defeated by Charles Martel was not to conquer the Frankish kingdom, but simply to pillage the wealthy monastery of St-Martin of Tours". Similarly, Tomaž Mastnak writes:

"Modern historians have constructed a myth presenting this victory as having saved Christian Europe from the Muslims. Edward Gibbon, for example, called Charles Martel the savior of Christendom and the battle near Poitiers an encounter that changed the history of the world. ... This myth has survived well into our own times. ... Contemporaries of the battle, however, did not overstate its significance. The continuators of Fredegar's chronicle, who probably wrote in the mid-eighth century, pictured the battle as just one of many military encounters between Christians and Saracens – moreover, as only one in a series of wars fought by Frankish princes for booty and territory. ... One of Fredegar's continuators presented the battle of [Tours-Poitiers] as what it really was: an episode in the struggle between Christian princes as the Carolingians strove to bring Aquitaine under their rule."

The historian Philip Khuri Hitti believes that "In reality, nothing was decided on the battlefield of Tours. The Moslem wave, already a thousand miles from its starting point in Gibraltar – to say nothing about its base in al-Qayrawan – had already spent itself and reached a natural limit."

The view that the battle has no great significance is perhaps best summarized by in Europe and Islam:

"Although prudence needs to be exercised in minimizing or 'demythologizing' the significance of the event, it is no longer thought by anyone to have been crucial. The 'myth' of that particular military engagement survives today as a media cliché, than which nothing is harder to eradicate. It is well known how the propaganda put about by the Franks and the papacy glorified the victory that took place on the road between Tours and Poitiers..."

In their introduction to The Reader's Companion to Military History Robert Cowley and Geoffrey Parker summarise this side of the modern view of the Battle of Tours by saying:

"The study of military history has undergone drastic changes in recent years. The old drums-and-bugles approach will no longer do. Factors such as economics, logistics, intelligence, and technology receive the attention once accorded solely to battles and campaigns and casualty counts. Words like 'strategy' and 'operations' have acquired meanings that might not have been recognizable a generation ago. Changing attitudes and new research have altered our views of what once seemed to matter most. For example, several of the battles that Edward Shepherd Creasy listed in his famous 1851 book The Fifteen Decisive Battles of the World rate hardly a mention here, and the confrontation between Muslims and Christians at Poitiers-Tours in 732, once considered a watershed event, has been downgraded to a raid in force."

title: Battle of Tours


text:
Xylem

Xylem is one of the two types of transport tissue in vascular plants, the other being phloem. The basic function of the xylem is to transport water from roots to stems and leaves, but it also transports nutrients. The word xylem is derived from the Ancient Greek word ξύλον (xylon), meaning "wood"; the best-known xylem tissue is wood, though it is found throughout a plant. The term was introduced by Carl Nägeli in 1858.

Structure
The most distinctive xylem cells are the long tracheary elements that transport water. Tracheids and vessel elements are distinguished by their shape; vessel elements are shorter, and are connected together into long tubes that are called vessels.

Xylem also contains two other type of cells: parenchyma and fibers.

Xylem can be found:
 * in vascular bundles, present in non-woody plants and non-woody parts of woody plants
 * in secondary xylem, laid down by a meristem called the vascular cambium in woody plants
 * as part of a stelar arrangement not divided into bundles, as in many ferns.

In transitional stages of plants with secondary growth, the first two categories are not mutually exclusive, although usually a vascular bundle will contain primary xylem only.

The branching pattern exhibited by xylem follows Murray's law.

Primary and secondary xylem
Primary xylem is formed during primary growth from procambium. It includes protoxylem and metaxylem. Metaxylem develops after the protoxylem but before secondary xylem. Metaxylem has wider vessels and tracheids than protoxylem.

Secondary xylem is formed during secondary growth from vascular cambium. Although secondary xylem is also found in members of the gymnosperm groups Gnetophyta and Ginkgophyta and to a lesser extent in members of the Cycadophyta, the two main groups in which secondary xylem can be found are:
 * 1) conifers (Coniferae): there are approximately 600 known species of conifers. All species have secondary xylem, which is relatively uniform in structure throughout this group. Many conifers become tall trees: the secondary xylem of such trees is used and marketed as softwood.
 * 2) angiosperms (Angiospermae): there are approximately 250,000 known species of angiosperms. Within this group secondary xylem is rare in the monocots. Many non-monocot angiosperms become trees, and the secondary xylem of these is used and marketed as hardwood.

Main function – upwards water transport
The xylem, vessels and tracheids of the roots, stems and leaves are interconnected to form a continuous system of water-conducting channels reaching all parts of the plants. The system transports water and soluble mineral nutrients from the roots throughout the plant. It is also used to replace water lost during transpiration and photosynthesis. Xylem sap consists mainly of water and inorganic ions, although it can also contain a number of organic chemicals as well. The transport is passive, not powered by energy spent by the tracheary elements themselves, which are dead by maturity and no longer have living contents. Transporting sap upwards becomes more difficult as the height of a plant increases and upwards transport of water by xylem is considered to limit the maximum height of trees. Three phenomena cause xylem sap to flow:
 * Pressure flow hypothesis: Sugars produced in the leaves and other green tissues are kept in the phloem system, creating a solute pressure differential versus the xylem system carrying a far lower load of solutes- water and minerals. The phloem pressure can rise to several MPa, far higher than atmospheric pressure. Selective inter-connection between these systems allows this high solute concentration in the phloem to draw xylem fluid upwards by negative pressure.
 * Transpirational pull: Similarly, the evaporation of water from the surfaces of mesophyll cells to the atmosphere also creates a negative pressure at the top of a plant. This causes millions of minute menisci to form in the mesophyll cell wall. The resulting surface tension causes a negative pressure or tension in the xylem that pulls the water from the roots and soil.
 * Root pressure: If the water potential of the root cells is more negative than that of the soil, usually due to high concentrations of solute, water can move by osmosis into the root from the soil. This causes a positive pressure that forces sap up the xylem towards the leaves. In some circumstances, the sap will be forced from the leaf through a hydathode in a phenomenon known as guttation. Root pressure is highest in the morning before the opening of stomata and allow transpiration to begin. Different plant species can have different root pressures even in a similar environment; examples include up to 145 kPa in Vitis riparia but around zero in Celastrus orbiculatus.

The primary force that creates the capillary action movement of water upwards in plants is the adhesion between the water and the surface of the xylem conduits. Capillary action provides the force that establishes an equilibrium configuration, balancing gravity. When transpiration removes water at the top, the flow is needed to return to the equilibrium.

Transpirational pull results from the evaporation of water from the surfaces of cells in the leaves. This evaporation causes the surface of the water to recess into the pores of the cell wall. By capillary action, the water forms concave menisci inside the pores. The high surface tension of water pulls the concavity outwards, generating enough force to lift water as high as a hundred meters from ground level to a tree's highest branches.

Transpirational pull requires that the vessels transporting the water be very small in diameter; otherwise, cavitation would break the water column. And as water evaporates from leaves, more is drawn up through the plant to replace it. When the water pressure within the xylem reaches extreme levels due to low water input from the roots (if, for example, the soil is dry), then the gases come out of solution and form a bubble – an embolism forms, which will spread quickly to other adjacent cells, unless bordered pits are present (these have a plug-like structure called a torus, that seals off the opening between adjacent cells and stops the embolism from spreading). Even after an embolism has occurred, plants are able to refill the xylem and restore the functionality.

Cohesion-tension theory
The cohesion-tension theory is a theory of intermolecular attraction that explains the process of water flow upwards (against the force of gravity) through the xylem of plants. It was proposed in 1894 by John Joly and Henry Horatio Dixon. Despite numerous objections, this is the most widely accepted theory for the transport of water through a plant's vascular system based on the classical research of Dixon-Joly (1894), Eugen Askenasy (1845–1903) (1895),  and Dixon (1914,1924).

Water is a polar molecule. When two water molecules approach one another, the slightly negatively charged oxygen atom of one forms a hydrogen bond with a slightly positively charged hydrogen atom in the other. This attractive force, along with other intermolecular forces, is one of the principal factors responsible for the occurrence of surface tension in liquid water. It also allows plants to draw water from the root through the xylem to the leaf.

Water is constantly lost through transpiration from the leaf. When one water molecule is lost another is pulled along by the processes of cohesion and tension. Transpiration pull, utilizing capillary action and the inherent surface tension of water, is the primary mechanism of water movement in plants. However, it is not the only mechanism involved. Any use of water in leaves forces water to move into them.

Transpiration in leaves creates tension (differential pressure) in the cell walls of mesophyll cells. Because of this tension, water is being pulled up from the roots into the leaves, helped by cohesion (the pull between individual water molecules, due to hydrogen bonds) and adhesion (the stickiness between water molecules and the hydrophilic cell walls of plants). This mechanism of water flow works because of water potential (water flows from high to low potential), and the rules of simple diffusion.

Over the past century, there has been a great deal of research regarding the mechanism of xylem sap transport; today, most plant scientists continue to agree that the cohesion-tension theory best explains this process, but multiforce theories that hypothesize several alternative mechanisms have been suggested, including longitudinal cellular and xylem osmotic pressure gradients, axial potential gradients in the vessels, and gel- and gas-bubble-supported interfacial gradients.

Measurement of pressure
Until recently, the differential pressure (suction) of transpirational pull could only be measured indirectly, by applying external pressure with a pressure bomb to counteract it. When the technology to perform direct measurements with a pressure probe was developed, there was initially some doubt about whether the classic theory was correct, because some workers were unable to demonstrate negative pressures. More recent measurements do tend to validate the classic theory, for the most part. Xylem transport is driven by a combination of transpirational pull from above and root pressure from below, which makes the interpretation of measurements more complicated.

Evolution
Xylem appeared early in the history of terrestrial plant life. Fossil plants with anatomically preserved xylem are known from the Silurian (more than 400 million years ago), and trace fossils resembling individual xylem cells may be found in earlier Ordovician rocks. The earliest true and recognizable xylem consists of tracheids with a helical-annular reinforcing layer added to the cell wall. This is the only type of xylem found in the earliest vascular plants, and this type of cell continues to be found in the protoxylem (first-formed xylem) of all living groups of vascular plants. Several groups of plants later developed pitted tracheid cells independently through convergent evolution. In living plants, pitted tracheids do not appear in development until the maturation of the metaxylem (following the protoxylem).

In most plants, pitted tracheids function as the primary transport cells. The other type of vascular element, found in angiosperms, is the vessel element. Vessel elements are joined end to end to form vessels in which water flows unimpeded, as in a pipe. The presence of xylem vessels (also called trachea ) is considered to be one of the key innovations that led to the success of the angiosperms. However, the occurrence of vessel elements is not restricted to angiosperms, and they are absent in some archaic or "basal" lineages of the angiosperms: (e.g., Amborellaceae, Tetracentraceae, Trochodendraceae, and Winteraceae), and their secondary xylem is described by Arthur Cronquist as "primitively vesselless". Cronquist considered the vessels of Gnetum to be convergent with those of angiosperms. Whether the absence of vessels in basal angiosperms is a primitive condition is contested, the alternative hypothesis states that vessel elements originated in a precursor to the angiosperms and were subsequently lost.

To photosynthesize, plants must absorb from the atmosphere. However, this comes at a price: while stomata are open to allow to enter, water can evaporate. Water is lost much faster than is absorbed, so plants need to replace it, and have developed systems to transport water from the moist soil to the site of photosynthesis. Early plants sucked water between the walls of their cells, then evolved the ability to control water loss (and acquisition) through the use of stomata. Specialized water transport tissues soon evolved in the form of hydroids, tracheids, then secondary xylem, followed by an endodermis and ultimately vessels.

The high levels of Silurian-Devonian times, when plants were first colonizing land, meant that the need for water was relatively low. As was withdrawn from the atmosphere by plants, more water was lost in its capture, and more elegant transport mechanisms evolved. As water transport mechanisms, and waterproof cuticles, evolved, plants could survive without being continually covered by a film of water. This transition from poikilohydry to homoiohydry opened up new potential for colonization. Plants then needed a robust internal structure that held long narrow channels for transporting water from the soil to all the different parts of the above-soil plant, especially to the parts where photosynthesis occurred.

During the Silurian, was readily available, so little water needed expending to acquire it. By the end of the Carboniferous, when levels had lowered to something approaching today's, around 17 times more water was lost per unit of  uptake. However, even in these "easy" early days, water was at a premium, and had to be transported to parts of the plant from the wet soil to avoid desiccation. This early water transport took advantage of the cohesion-tension mechanism inherent in water. Water has a tendency to diffuse to areas that are drier, and this process is accelerated when water can be wicked along a fabric with small spaces. In small passages, such as that between the plant cell walls (or in tracheids), a column of water behaves like rubber – when molecules evaporate from one end, they pull the molecules behind them along the channels. Therefore, transpiration alone provided the driving force for water transport in early plants. However, without dedicated transport vessels, the cohesion-tension mechanism cannot transport water more than about 2 cm, severely limiting the size of the earliest plants. This process demands a steady supply of water from one end, to maintain the chains; to avoid exhausting it, plants developed a waterproof cuticle. Early cuticle may not have had pores but did not cover the entire plant surface, so that gas exchange could continue. However, dehydration at times was inevitable; early plants cope with this by having a lot of water stored between their cell walls, and when it comes to it sticking out the tough times by putting life "on hold" until more water is supplied.

To be free from the constraints of small size and constant moisture that the parenchymatic transport system inflicted, plants needed a more efficient water transport system. During the early Silurian, they developed specialized cells, which were lignified (or bore similar chemical compounds) to avoid implosion; this process coincided with cell death, allowing their innards to be emptied and water to be passed through them. These wider, dead, empty cells were a million times more conductive than the inter-cell method, giving the potential for transport over longer distances, and higher diffusion rates.

The earliest macrofossils to bear water-transport tubes are Silurian plants placed in the genus Cooksonia. The early Devonian pretracheophytes Aglaophyton and Horneophyton have structures very similar to the hydroids of modern mosses. Plants continued to innovate new ways of reducing the resistance to flow within their cells, thereby increasing the efficiency of their water transport. Bands on the walls of tubes, in fact apparent from the early Silurian onwards, are an early improvisation to aid the easy flow of water. Banded tubes, as well as tubes with pitted ornamentation on their walls, were lignified and, when they form single celled conduits, are considered to be tracheids. These, the "next generation" of transport cell design, have a more rigid structure than hydroids, allowing them to cope with higher levels of water pressure. Tracheids may have a single evolutionary origin, possibly within the hornworts, uniting all tracheophytes (but they may have evolved more than once).

Water transport requires regulation, and dynamic control is provided by stomata. By adjusting the amount of gas exchange, they can restrict the amount of water lost through transpiration. This is an important role where water supply is not constant, and indeed stomata appear to have evolved before tracheids, being present in the non-vascular hornworts.

An endodermis probably evolved during the Silu-Devonian, but the first fossil evidence for such a structure is Carboniferous. This structure in the roots covers the water transport tissue and regulates ion exchange (and prevents unwanted pathogens etc. from entering the water transport system). The endodermis can also provide an upwards pressure, forcing water out of the roots when transpiration is not enough of a driver.

Once plants had evolved this level of controlled water transport, they were truly homoiohydric, able to extract water from their environment through root-like organs rather than relying on a film of surface moisture, enabling them to grow to much greater size. As a result of their independence from their surroundings, they lost their ability to survive desiccation – a costly trait to retain.

During the Devonian, maximum xylem diameter increased with time, with the minimum diameter remaining pretty constant. By the middle Devonian, the tracheid diameter of some plant lineages (Zosterophyllophytes) had plateaued. Wider tracheids allow water to be transported faster, but the overall transport rate depends also on the overall cross-sectional area of the xylem bundle itself. The increase in vascular bundle thickness further seems to correlate with the width of plant axes, and plant height; it is also closely related to the appearance of leaves and increased stomatal density, both of which would increase the demand for water.

While wider tracheids with robust walls make it possible to achieve higher water transport tensions, this increases the likelihood of cavitation. Cavitation occurs when a bubble of air forms within a vessel, breaking the bonds between chains of water molecules and preventing them from pulling more water up with their cohesive tension. A tracheid, once cavitated, cannot have its embolism removed and return to service (except in a few advanced angiosperms which have developed a mechanism of doing so). Therefore, it is well worth plants' while to avoid cavitation occurring. For this reason, pits in tracheid walls have very small diameters, to prevent air entering and allowing bubbles to nucleate. Freeze-thaw cycles are a major cause of cavitation. Damage to a tracheid's wall almost inevitably leads to air leaking in and cavitation, hence the importance of many tracheids working in parallel.

Once cavitation has occurred, plants have a range of mechanisms to contain the damage. Small pits link adjacent conduits to allow fluid to flow between them, but not air – although these pits, which prevent the spread of embolism, are also a major cause of them. These pitted surfaces further reduce the flow of water through the xylem by as much as 30%. The diversification of xylem strand shapes with tracheid network topologies increasingly resistant to the spread of embolism likely facilitated increases in plant size and the colonization of drier habitats during the Devonian radiation. Conifers, by the Jurassic, developed bordered pits had valve-like structures to isolate cavitated elements. These torus-margo structures have an impermeable disc (torus) suspended by a permeable membrane (margo) between two adjacent pores. When a tracheid on one side depressurizes, the disc is sucked into the pore on that side, and blocks further flow. Other plants simply tolerate cavitation. For instance, oaks grow a ring of wide vessels at the start of each spring, none of which survive the winter frosts. Maples use root pressure each spring to force sap upwards from the roots, squeezing out any air bubbles.

Growing to height also employed another trait of tracheids – the support offered by their lignified walls. Defunct tracheids were retained to form a strong, woody stem, produced in most instances by a secondary xylem. However, in early plants, tracheids were too mechanically vulnerable, and retained a central position, with a layer of tough sclerenchyma on the outer rim of the stems. Even when tracheids do take a structural role, they are supported by sclerenchymatic tissue.

Tracheids end with walls, which impose a great deal of resistance on flow; vessel members have perforated end walls, and are arranged in series to operate as if they were one continuous vessel. The function of end walls, which were the default state in the Devonian, was probably to avoid embolisms. An embolism is where an air bubble is created in a tracheid. This may happen as a result of freezing, or by gases dissolving out of solution. Once an embolism is formed, it usually cannot be removed (but see later); the affected cell cannot pull water up, and is rendered useless.

End walls excluded, the tracheids of prevascular plants were able to operate under the same hydraulic conductivity as those of the first vascular plant, Cooksonia.

The size of tracheids is limited as they comprise a single cell; this limits their length, which in turn limits their maximum useful diameter to 80 μm. Conductivity grows with the fourth power of diameter, so increased diameter has huge rewards; vessel elements, consisting of a number of cells, joined at their ends, overcame this limit and allowed larger tubes to form, reaching diameters of up to 500 μm, and lengths of up to 10 m.

Vessels first evolved during the dry, low periods of the late Permian, in the horsetails, ferns and Selaginellales independently, and later appeared in the mid Cretaceous in angiosperms and gnetophytes. Vessels allow the same cross-sectional area of wood to transport around a hundred times more water than tracheids! This allowed plants to fill more of their stems with structural fibers, and also opened a new niche to vines, which could transport water without being as thick as the tree they grew on. Despite these advantages, tracheid-based wood is a lot lighter, thus cheaper to make, as vessels need to be much more reinforced to avoid cavitation.

Development
Xylem development can be described by four terms: centrarch, exarch, endarch and mesarch. As it develops in young plants, its nature changes from protoxylem to metaxylem (i.e. from first xylem to after xylem). The patterns in which protoxylem and metaxylem are arranged is important in the study of plant morphology.

Protoxylem and metaxylem
As a young vascular plant grows, one or more strands of primary xylem form in its stems and roots. The first xylem to develop is called 'protoxylem'. In appearance protoxylem is usually distinguished by narrower vessels formed of smaller cells. Some of these cells have walls which contain thickenings in the form of rings or helices. Functionally, protoxylem can extend: the cells are able to grow in size and develop while a stem or root is elongating. Later, 'metaxylem' develops in the strands of xylem. Metaxylem vessels and cells are usually larger; the cells have thickenings which are typically either in the form of ladderlike transverse bars (scalariform) or continuous sheets except for holes or pits (pitted). Functionally, metaxylem completes its development after elongation ceases when the cells no longer need to grow in size.

Patterns of protoxylem and metaxylem
There are four main patterns to the arrangement of protoxylem and metaxylem in stems and roots.

 * Centrarch refers to the case in which the primary xylem forms a single cylinder in the center of the stem and develops from the center outwards. The protoxylem is thus found in the central core and the metaxylem in a cylinder around it. This pattern was common in early land plants, such as "rhyniophytes", but is not present in any living plants.

The other three terms are used where there is more than one strand of primary xylem.

 * Exarch is used when there is more than one strand of primary xylem in a stem or root, and the xylem develops from the outside inwards towards the center, i.e. centripetally. The metaxylem is thus closest to the center of the stem or root and the protoxylem closest to the periphery. The roots of vascular plants are normally considered to have exarch development.
 * Endarch is used when there is more than one strand of primary xylem in a stem or root, and the xylem develops from the inside outwards towards the periphery, i.e. centrifugally. The protoxylem is thus closest to the center of the stem or root and the metaxylem closest to the periphery. The stems of seed plants typically have endarch development.
 * Mesarch is used when there is more than one strand of primary xylem in a stem or root, and the xylem develops from the middle of a strand in both directions. The metaxylem is thus on both the peripheral and central sides of the strand with the protoxylem between the metaxylem (possibly surrounded by it). The leaves and stems of many ferns have mesarch development.

History
In his book De plantis libri XVI (On Plants, in 16 books) (1583), the Italian physician and botanist Andrea Cesalpino proposed that plants draw water from soil not by magnetism (ut magnes ferrum trahit, as magnetic iron attracts) nor by suction (vacuum), but by absorption, as occurs in the case of linen, sponges, or powders. The Italian biologist Marcello Malpighi was the first person to describe and illustrate xylem vessels, which he did in his book Anatome plantarum ... (1675). Although Malpighi believed that xylem contained only air, the British physician and botanist Nehemiah Grew, who was Malpighi's contemporary, believed that sap ascended both through the bark and through the xylem. However, according to Grew, capillary action in the xylem would raise the sap by only a few inches; in order to raise the sap to the top of a tree, Grew proposed that the parenchymal cells become turgid and thereby not only squeeze the sap in the tracheids but force some sap from the parenchyma into the tracheids. In 1727, English clergyman and botanist Stephen Hales showed that transpiration by a plant's leaves causes water to move through its xylem. By 1891, the Polish-German botanist Eduard Strasburger had shown that the transport of water in plants did not require the xylem cells to be alive.

General references

 * is the main source used for the paragraph on recent research.
 * is the first published independent test showing the Scholander bomb actually does measure the tension in the xylem.
 * is the second published independent test showing the Scholander bomb actually does measure the tension in the xylem.
 * recent update of the classic book on xylem transport by the late Martin Zimmermann
 * recent update of the classic book on xylem transport by the late Martin Zimmermann
 * recent update of the classic book on xylem transport by the late Martin Zimmermann
 * recent update of the classic book on xylem transport by the late Martin Zimmermann

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

title: Xylem


text:
Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.

Statement
Definition. Let $$(X, d)$$ be a metric space. Then a map $$T : X \to X$$ is called a contraction mapping on X if there exists $$q \in [0, 1)$$ such that
 * $$d(T(x),T(y)) \le q d(x,y)$$

for all $$x, y \in X.$$

Banach fixed-point theorem. Let $$(X, d)$$ be a non-empty complete metric space with a contraction mapping $$T : X \to X.$$ Then T admits a unique fixed-point $$x^*$$ in X (i.e. $$T(x^*) = x^*$$). Furthermore, $$x^*$$ can be found as follows: start with an arbitrary element $$x_0 \in X$$ and define a sequence $$(x_n)_{n\in\mathbb N}$$ by $$x_n = T(x_{n-1})$$ for $$n \geq 1.$$ Then $$\lim_{n \to \infty} x_n = x^*$$.

Remark 1. The following inequalities are equivalent and describe the speed of convergence:

\begin{align} d(x^*, x_n) & \leq \frac{q^n}{1-q} d(x_1,x_0), \\[5pt] d(x^*, x_{n+1}) & \leq \frac{q}{1-q} d(x_{n+1},x_n), \\[5pt] d(x^*, x_{n+1}) & \leq q d(x^*,x_n). \end{align} $$

Any such value of q is called a Lipschitz constant for $$T$$, and the smallest one is sometimes called "the best Lipschitz constant" of $$T$$.

Remark 2. $$d(T(x),T(y))<d(x,y)$$ for all $$x \neq y$$ is in general not enough to ensure the existence of a fixed point, as is shown by the map
 * $$T : [1,\infty) \to [1,\infty), \,\, T(x)=x+\tfrac{1}{x}\,,$$

which lacks a fixed point. However, if $$X$$ is compact, then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of $$d(x,T(x))$$, indeed, a minimizer exists by compactness, and has to be a fixed point of $$T.$$ It then easily follows that the fixed point is the limit of any sequence of iterations of $$T.$$

Remark 3. When using the theorem in practice, the most difficult part is typically to define $$X$$ properly so that $$T(X) \subseteq X.$$

Proof
Let $$x_0 \in X$$ be arbitrary and define a sequence $$(x_n)_{n\in\mathbb N}$$ by setting $$x_n = T(x_{n-1})$$. We first note that for all $$n \in \N,$$ we have the inequality

 * $$d(x_{n+1}, x_n) \le q^n d(x_1, x_0).$$

This follows by induction on n, using the fact that T is a contraction mapping. Then we can show that $$(x_n)_{n\in\mathbb N}$$ is a Cauchy sequence. In particular, let $$m, n \in \N$$ such that $$m > n $$:

 * $$\begin{align}

d(x_m, x_n) & \leq d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + \cdots + d(x_{n+1}, x_n) \\[5pt] & \leq q^{m-1}d(x_1, x_0) + q^{m-2}d(x_1, x_0) + \cdots + q^nd(x_1, x_0) \\[5pt] & = q^n d(x_1, x_0) \sum_{k=0}^{m-n-1} q^k \\[5pt] & \leq q^n d(x_1, x_0) \sum_{k=0}^\infty q^k \\[5pt] & = q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ). \end{align}$$

Let ε > 0 be arbitrary. Since $$q \in [0,1)$$, we can find a large $$N \in \N$$ so that

 * $$q^N < \frac{\varepsilon(1-q)}{d(x_1, x_0)}.$$

Therefore, by choosing $$m$$ and $$n$$ greater than $$N$$ we may write:

 * $$d(x_m, x_n) \leq q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ) < \left (\frac{\varepsilon(1-q)}{d(x_1, x_0)} \right ) d(x_1, x_0) \left ( \frac{1}{1-q} \right ) = \varepsilon.$$

This proves that the sequence $$(x_n)_{n\in\mathbb N}$$ is Cauchy. By completeness of (X,d), the sequence has a limit $$x^* \in X.$$ Furthermore, $$x^*$$ must be a fixed point of T:

 * $$x^*=\lim_{n\to\infty} x_n = \lim_{n\to\infty} T(x_{n-1}) = T\left(\lim_{n\to\infty} x_{n-1} \right) = T(x^*). $$

As a contraction mapping, T is continuous, so bringing the limit inside T was justified. Lastly, T cannot have more than one fixed point in (X,d), since any pair of distinct fixed points p1 and p2 would contradict the contraction of T:

 * $$ d(T(p_1),T(p_2)) = d(p_1,p_2) > q d(p_1, p_2).$$

Applications

 * A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator on the space of continuous functions under the uniform norm. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.
 * One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach space E; let I : Ω → E denote the identity (inclusion) map and let g : Ω → E be a Lipschitz map of constant k < 1. Then
 * 1) Ω′ := (I + g)(Ω) is an open subset of E: precisely, for any x in Ω such that B(x, r) ⊂ Ω one has B((I + g)(x), r(1 − k)) ⊂ Ω′;
 * 2) I + g : Ω → Ω′ is a bi-Lipschitz homeomorphism;
 * precisely, (I + g)−1 is still of the form I + h : Ω → Ω′ with h a Lipschitz map of constant k/(1 − k). A direct consequence of this result yields the proof of the inverse function theorem.

 * It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third-order method.
 * It can be used to prove existence and uniqueness of solutions to integral equations.
 * It can be used to give a proof to the Nash embedding theorem.
 * It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of reinforcement learning.
 * It can be used to prove existence and uniqueness of an equilibrium in Cournot competition, and other dynamic economic models.

Converses
Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959:

Let f : X → X be a map of an abstract set such that each iterate fn has a unique fixed point. Let $$q \in (0, 1),$$ then there exists a complete metric on X such that f is contractive, and q is the contraction constant.

Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if $$f : X \to X$$ is a map on a T1 topological space with a unique fixed point a, such that for each $$x \in X$$ we have fn(x) → a, then there already exists a metric on X with respect to which f satisfies the conditions of the Banach contraction principle with contraction constant 1/2. In this case the metric is in fact an ultrametric.

Generalizations
There are a number of generalizations (some of which are immediate corollaries).

Let T : X → X be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:
 * Assume that some iterate Tn of T is a contraction. Then T has a unique fixed point.
 * Assume that for each n, there exist cn such that d(Tn(x), Tn(y)) ≤ cnd(x, y) for all x and y, and that
 * $$\sum\nolimits_n c_n <\infty.$$
 * Then T has a unique fixed point.

In applications, the existence and uniqueness of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map T a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems in infinite-dimensional spaces for generalizations.

A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric. Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.

Example
Banach theorem allows for example fast and accurate calculation of the $\pi$ number using the trigonometric functions which numerically are the power Taylor series.

Because $$\sin(\pi)=0$$ and the π is the fixed point of for example the function $$f(x)=\sin(x)+x$$

i.e.

 * $$f(\pi)=\pi$$

and also the function $$f$$ is around π the contraction mapping from the obvious reasons because its derivative in π vanishes therefore π can be obtained from the infinite superposition for example for the argument value 3:

 * $$\pi=f(f(f(\cdots f(3)\cdots))))$$

Already the triple superposition of this function at $$3$$ gives π with accuracy to 33 digits:

 * $$f(f(f(3)))=3.141592653589793238462643383279502\ldots\,.$$

title: Banach fixed-point theorem


text:
Die Fledermaus

Die Fledermaus (, The Flittermouse or The Bat, sometimes called The Revenge of the Bat) is an operetta composed by Johann Strauss II to a German libretto by Karl Haffner and Richard Genée, which premiered in 1874.

Background
The original literary source for Die Fledermaus was Das Gefängnis (The Prison), a farce by German playwright Julius Roderich Benedix that premiered in Berlin in 1851. On 10 September 1872, a three-act French vaudeville play by Henri Meilhac and Ludovic Halévy, Le Réveillon, loosely based on the Benedix farce, opened at the Théâtre du Palais-Royal. Meilhac and Halévy had provided several successful libretti for Offenbach and Le Réveillon later formed the basis for the 1926 silent film So This Is Paris, directed by Ernst Lubitsch.

Meilhac and Halévy's play was soon translated into German by Karl Haffner (1804–1876), at the instigation of Max Steiner, as a non-musical play for production in Vienna. The French custom of a New Year's Eve réveillon, or supper party, was not considered to provide a suitable setting for the Viennese theatre, so it was decided to substitute a ball for the réveillon. Haffner's translation was then passed to the playwright and composer Richard Genée, who had provided some of the lyrics for Strauss's Der Karneval in Rom the year before, and he completed the libretto.

Performance history
The operetta premiered on 5 April 1874 at the Theater an der Wien in Vienna and has been part of the regular repertoire ever since.

It was performed in New York under Rudolf Bial at the Stadt Theatre on 21 November 1874. The German première took place at Munich's Gärtnerplatztheater in 1875. Die Fledermaus was sung in English at London's Alhambra Theatre on 18 December 1876, with its score modified by Hamilton Clarke. When the operetta came to Paris in 1877 at the Théâtre de la Renaissance, as La Tzigane, with Ismaël and Zulma Bouffar in the cast, it was not a success; only in 1904, with Meilhac and Halevy's original roles names and the words adapted by Paul Ferrier to the music (with Max Dearly and Ève Lavallière in the cast) did it find success in Paris and enter the repertoire there.

The first London performance in German did not take place until 1895. According to the archivist of the Royal Opera House, Covent Garden, "Twenty years after its production as a lyric opera in Vienna, [composer and conductor Gustav] Mahler raised the artistic status of Strauss's work by producing it at the Hamburg Opera House [...] all the leading opera houses in Europe, notably Vienna and Munich, have brightened their regular repertoire by including it for occasional performance."

The role of Eisenstein was originally written for a tenor but is now frequently sung by a baritone. The role of Orlofsky is a trouser role, usually performed by a mezzo-soprano, sometimes by a countertenor and occasionally – an octave lower – by a tenor.

The party of act 2 allows productions to insert a variety of additional entertainment acts, such as music, comedy, or dance. The lengthy drunken soliloquy by Frosch (a comedy speaking role) in act 3 also permits variety in performance.

Act 1
Eisenstein's apartment

Gabriel von Eisenstein, a Viennese man-about-town, has been sentenced to eight days in prison for insulting an official, partially due to the incompetence of his attorney, Dr. Blind. Adele, Eisenstein's maid, receives a forged letter, allegedly from her sister who is in the company of the ballet, but actually written by Falke, inviting her to Prince Orlofsky's ball. She pretends the letter says that her aunt is very sick, and asks her mistress Rosalinde (Eisenstein's wife) for an evening off ("Da schreibt meine Schwester Ida"/"My sister Ida writes to me"). Falke, Eisenstein's friend, arrives to invite him to the ball (Duet: "Komm mit mir zum Souper"/"Come with me to the supper"). Together, they recall a practical joke which Eisenstein played on Falke a few years ago, for which Falke is secretly planning a light-hearted revenge in kind. Eisenstein bids farewell to Adele and his wife Rosalinde, pretending he is going to prison (Trio: "O Gott, wie rührt mich dies!"/"Oh dear, oh dear, how sorry I am") but really intending to postpone jail for one day and have fun at the ball.

After Eisenstein leaves, Rosalinde is visited by her former lover, the singing teacher Alfred, who serenades her ("Täubchen, das entflattert ist"/"Dove that has escaped"). Frank, the governor of the prison, arrives to take Eisenstein to jail, and finds Alfred instead. In order not to compromise Rosalinde, Alfred agrees to pretend to be Eisenstein and to accompany Frank. (Finale, drinking song: "Glücklich ist, wer vergisst"/"Happy is he who forgets" followed by Rosalinde's defence when Frank arrives: "Mit mir so spät im tête-à-tête"/"In tête-à-tête with me so late," and Frank's invitation: "Mein schönes, großes Vogelhaus"/"My beautiful, large bird-cage.")

Act 2
A summer house in the Villa Orlofsky

It transpires that Falke, with Prince Orlofsky's permission, is using the ball as a way of getting revenge on Eisenstein. Some time before, after a costume-party, Eisenstein had abandoned Falke, very drunk and dressed in a bat-costume, in the center of town, exposing him to ridicule the next day. As part of his scheme, Falke has invited Frank, Adele, and Rosalinde to come the ball, all concealing their identities as well. Rosalinde pretends to be a masked Hungarian countess, Eisenstein goes by the name "Marquis Renard," Frank is "Chevalier Chagrin," and Adele, who has borrowed one of Rosalinde's dresses without permission, pretends she is an actress.

The ball is in progress (Chorus: "Ein Souper heut' uns winkt"/"A supper is before us") and the Prince welcomes his guests ("Ich lade gern mir Gäste ein"/"I love to invite my friends"). Eisenstein is introduced to Adele, but is confused as to who she really is because of her striking resemblance to his maid. ("Mein Herr Marquis"/"My lord marquis," sometimes referred to as "Adele's Laughing Song"). Frank arrives. He and Eisenstein, who are both posing as Frenchmen, attempt to conceal their identities by repeating common French phrases to each other, to Orlofsky's great amusement. Since neither actually knows French, both are fooled. As the party progresses, they both experience alcohol-induced good-feeling and manly camaraderie for each other.

Then Falke introduces the masked Rosalinde to the company. She convinces everyone that she is Hungarian by singing the "Czardas", a sentimental dancing-song ("Klänge der Heimat"/"Sounds from home"). During an amorous tête-à-tête, Eisenstein tries unsuccessfully to persuade the mystery-woman to unmask. She succeeds in extracting a valuable watch from her husband's pocket, something which she can use in the future as evidence of his impropriety. (Watch duet: "Dieser Anstand, so manierlich"/"Her bearing, so well-mannered"). In a rousing finale, Orlofsky makes a toast to champagne, and the company celebrates (The Champagne song: "Im Feuerstrom der Reben"/"In the fire stream of the grape"; followed by the canon: "Brüderlein, Brüderlein und Schwesterlein"/"Brothers, brothers and sisters" and the waltz finale, "Ha, welch ein Fest, welche Nacht voll Freud'!"/"Ha, what joy, what a night of delight.") Eisenstein and Frank dash off as the clock strikes six in the morning.

(Note: The "Champagne song", which is sung by the entire ensemble, should not be confused with the baritone aria "Fin ch' han dal vino" from Don Giovanni, which is often called the "Champagne aria".)

Act 3
In the prison offices of Warden Frank

The next morning they all find themselves at the prison where the confusion increases and is compounded by the jailer, Frosch, who has profited by Warden Frank's absence to become gloriously drunk. Alfred, still in jail in Eisenstein's place, irritates the other prisoners by singing operatic arias.

Adele arrives to ask the Chevalier Chagrin (actually Frank) to sponsor her career as an actress, but Frank is not wealthy enough to do this (Melodrama; Couplet of Adele: "Spiel' ich die Unschuld vom Lande"/"If I play the innocent peasant maid"). Meanwhile, Alfred asks Frosch to summon Dr. Blind to help get him released; Frank agrees to allow this and Dr. Blind arrives. Eisenstein enters and says he has come to serve his sentence. He is surprised when Frank tells him that his cell is already occupied by a man who claims to be Eisenstein and whom Frank had arrested in Eisenstein's apartment. Frank further tells Eisenstein that the man he arrested was singing amorous songs to Rosalinde at the time of his arrest, and warmly kissed her goodbye. Enraged, Eisenstein takes Dr. Blind's wig and glasses in order to disguise himself and confront the impersonator Alfred, who Eisenstein now believes has cuckolded him. Rosalinde enters. Eisenstein takes off his disguise and accuses her of being unfaithful to him with Alfred. Eisenstein, Rosalinde, and Alfred sing a trio in which Eisenstein angrily claims the right of vengeance (Trio: "Ja, ich bin's, den ihr betrogen...Ra-ra-ra-ra-Rache will ich!"/"I'm the one who was mistreated....Ve-ve-ve-ve-vengeance is mine!"). However, Rosalinde produces his watch, and he realizes that the Hungarian mystery-woman he tried to seduce at Orlofsky's party was actually Rosalinde in disguise and that he, not she, is at fault.

Falke enters with all the guests from the party and explains that the whole thing was payback for Eisenstein's practical joke on him three years before. Eisenstein is delighted by the prank, and he begs Rosalinde to forgive him for his attempted infidelity. Rosalinde refuses at first and threatens to divorce him, but Eisenstein tells her that his misbehavior was caused by the champagne. She accepts this explanation and immediately forgives him unconditionally. Orlofsky promises to finance Adele's acting career, and the company joyfully reprises the "Champagne song" from act 2.

Stage adaptations
Strauss's music has been adopted for at least two musical comedies:
 * Nightbirds with lyrics by Gladys Unger, opened at the Lyric Theatre, London 30 December 1911
 * Gay Rosalinda, adaptation by Erich Wolfgang Korngold, produced at the Palace Theatre, London in 1945

Film adaptations
Die Fledermaus has been adapted numerous times for the cinema and for TV:

title: Die Fledermaus


text:
Entropy coding

In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have an expected code length greater than or equal to the entropy of the source.

More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies $$\operatorname E_{x\sim P}[\ell(d(x))] \geq \operatorname E_{x\sim P}[-\log_b(P(x))]$$, where $$\ell$$ is the number of symbols in a code word, $$d$$ is the coding function, $$b$$ is the number of symbols used to make output codes and $$P$$ is the probability of the source symbol. An entropy coding attempts to approach this lower bound.

Two of the most common entropy coding techniques are Huffman coding and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code may be useful. These static codes include universal codes (such as Elias gamma coding or Fibonacci coding) and Golomb codes (such as unary coding or Rice coding).

Since 2014, data compressors have started using the asymmetric numeral systems family of entropy coding techniques, which allows combination of the compression ratio of arithmetic coding with a processing cost similar to Huffman coding.

Entropy as a measure of similarity
Besides using entropy coding as a way to compress digital data, an entropy encoder can also be used to measure the amount of similarity between streams of data and already existing classes of data. This is done by generating an entropy coder/compressor for each class of data; unknown data is then classified by feeding the uncompressed data to each compressor and seeing which compressor yields the highest compression. The coder with the best compression is probably the coder trained on the data that was most similar to the unknown data.

title: Entropy coding