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Arago spot

In optics, the Arago spot, Poisson spot, or Fresnel spot is a bright point that appears at the center of a circular object's shadow due to Fresnel diffraction. This spot played an important role in the discovery of the wave nature of light and is a common way to demonstrate that light behaves as a wave.

The basic experimental setup requires a point source, such as an illuminated pinhole or a diverging laser beam. The dimensions of the setup must comply with the requirements for Fresnel diffraction. Namely, the Fresnel number must satisfy $$F = \frac{d^2}{\ell \lambda} \gtrsim 1,$$ where
 * $d$ is the diameter of the circular object,
 * $ℓ$ is the distance between the object and the screen, and
 * $λ$ is the wavelength of the source.

Finally, the edge of the circular object must be sufficiently smooth.

These conditions together explain why the bright spot is not encountered in everyday life. However, with the laser sources available today, it is undemanding to perform an Arago-spot experiment.

In astronomy, the Arago spot can also be observed in the strongly defocussed image of a star in a Newtonian telescope. There, the star provides an almost ideal point source at infinity, and the secondary mirror of the telescope constitutes the circular obstacle.

When light shines on the circular obstacle, Huygens' principle says that every point in the plane of the obstacle acts as a new point source of light. The light coming from points on the circumference of the obstacle and going to the center of the shadow travels exactly the same distance, so all the light passing close by the object arrives at the screen in phase and constructively interferes. This results in a bright spot at the shadow's center, where geometrical optics and particle theories of light predict that there should be no light at all.

History
At the beginning of the 19th century, the idea that light does not simply propagate along straight lines gained traction. Thomas Young published his double-slit experiment in 1807. The original Arago spot experiment was carried out a decade later and was the deciding experiment on the question of whether light is a particle or a wave. It is thus an example of an experimentum crucis.

At that time, many favored Isaac Newton's corpuscular theory of light, among them the theoretician Siméon Denis Poisson. In 1818 the French Academy of Sciences launched a competition to explain the properties of light, where Poisson was one of the members of the judging committee. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new wave theory of light.

Poisson studied Fresnel's theory in detail and, being a supporter of the particle theory of light, looked for a way to prove it wrong. Poisson thought that he had found a flaw when he argued that a consequence of Fresnel's theory was that there would exist an on-axis bright spot in the shadow of a circular obstacle, where there should be complete darkness according to the particle theory of light. This prediction was seen as an absurd consequence of the wave theory, and the failure of that prediction should be a strong argument to reject Fresnel's theory.

However, the head of the committee, Dominique-François-Jean Arago, decided to actually perform the experiment. He molded a 2 mm metallic disk to a glass plate with wax. He succeeded in observing the predicted spot, which convinced most scientists of the wave nature of light and gave Fresnel the win.

Arago later noted that the phenomenon (later known as "Poisson's spot" or the "spot of Arago") had already been observed by Delisle and Maraldi a century earlier.

Although Arago's experimental result was overwhelming evidence in favor of the wave theory, a century later, in conjunction with the birth of quantum mechanics (and first suggested in one of Albert Einstein's Annus Mirabilis papers), it became understood that light (as well as all forms of matter and energy) must be described as both a particle and a wave (wave–particle duality). However the particle associated with electromagnetic waves, the photon, has nothing in common with the particles imagined in the corpuscular theory that had been dominant before the rise of the wave theory and Arago's powerful demonstration. Before the advent of quantum theory in the late 1920s, only the wave nature of light could explain phenomena such as diffraction and interference. Today it is known that a diffraction pattern appears through the mosaic-like buildup of bright spots caused by single photons, as predicted by Dirac's quantum theory. With increasing light intensity the bright dots in the mosaic diffraction pattern just assemble faster. In contrast, the wave theory predicts the formation of an extended continuous pattern whose overall brightness increases with light intensity.

Theory

At the heart of Fresnel's wave theory is the Huygens–Fresnel principle, which states that every unobstructed point of a wavefront becomes the source of a secondary spherical wavelet and that the amplitude of the optical field E at a point on the screen is given by the superposition of all those secondary wavelets taking into account their relative phases. This means that the field at a point P1 on the screen is given by a surface integral: $$U(P_1) = \frac{A e^{\mathbf{i} k r_0}}{r_0} \int_S \frac{e^{\mathbf{i} k r_1}}{r_1} K(\chi) \, dS,$$ where the inclination factor $$K(\chi)$$ which ensures that the secondary wavelets do not propagate backwards is given by $$K(\chi) = \frac{\mathbf{i}}{2 \lambda} (1 + \cos(\chi))$$ and
 * A is the amplitude of the source wave
 * $k = \frac{2\pi}{\lambda}$ is the wavenumber
 * S is the unobstructed surface.

The first term outside of the integral represents the oscillations from the source wave at a distance r0. Similarly, the term inside the integral represents the oscillations from the secondary wavelets at distances r1.

In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the near-field diffraction regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances g = P0C and b = CP1). Going to polar coordinates then yields the integral for a circular object of radius a (see for example Born and Wolf ): $$U(P_1) = - \frac{\mathbf{i}}{\lambda} \frac{A e^{\mathbf{i} k (g + b)}}{g b} 2\pi \int_a^\infty e^{\mathbf{i} k \frac{1}{2} \left(\frac{1}{g} + \frac{1}{b}\right) r^2} r \, dr.$$

This integral can be solved numerically (see below). If g is large and b is small so that the angle $$\chi$$ is not negligible one can write the integral for the on-axis case (P1 is at the center of the shadow) as (see ): $$U(P_1) = \frac{A e^{\mathbf{i} k g}}{g} \frac{b}{\sqrt{b^2 + a^2}} e^{\mathbf{i} k \sqrt{b^2 + a^2}}.$$

The source intensity, which is the square of the field amplitude, is $I_0 = \left|\frac{1}{g} A e^{\mathbf{i} k g}\right|^2$ and the intensity at the screen $$I = \left| U(P_1) \right|^2$$. The on-axis intensity as a function of the distance b is hence given by: $$I = \frac{b^2}{b^2 + a^2} I_0.$$

This shows that the on-axis intensity at distances b much greater than the diameter of the circular obstacle is the same as the source intensity, as if the circular object was not present at all. However at larger distances b, it turns out that the size of the bright spot (as can be seen in the simulations below where b/a is increased in successive images) is larger therefore making the spot easier to discern.

Calculation of diffraction images
To calculate the full diffraction image that is visible on the screen one has to consider the surface integral of the previous section. One cannot exploit circular symmetry anymore, since the line between the source and an arbitrary point on the screen does not pass through the center of the circular object. With the aperture function $$g(r,\theta)$$ which is 1 for transparent parts of the object plane and 0 otherwise (i.e. It is 0 if the direct line between source and the point on the screen passes through the blocking circular object.) the integral that needs to be solved is given by: $$U(P_1) \propto \int_0^{2\pi} \int_0^\infty g(r,\theta) e^{\frac{\mathbf{i} \pi \rho^2}{\lambda} \left( \frac{1}{g} + \frac{1}{b} \right)} \rho \, d\rho \, d\theta.$$

Numerical calculation of the integral using the trapezoidal rule or Simpson's rule is not efficient and becomes numerically unstable especially for configurations with large Fresnel number. However, it is possible to solve the radial part of the integral so that only the integration over the azimuth angle remains to be done numerically. For a particular angle one must solve the line integral for the ray with origin at the intersection point of the line P0P1 with the circular object plane. The contribution for a particular ray with azimuth angle $$\theta_1$$ and passing a transparent part of the object plane from $$r = s$$ to $$r = t$$ is: $$R(\theta_1) \propto e^{\frac{\pi}{2} \mathbf{i} s^2} - e^{\frac{\pi}{2} \mathbf{i} t^2}.$$

So for each angle one has to compute the intersection point(s) of the ray with the circular object and then sum the contributions $$I(\theta_1)$$ for a certain number of angles between 0 and $$2\pi$$. Results of such a calculation are shown in the following images.

The images are simulations of the Arago spot in the shadow of discs of diameter 4 mm, 2 mm, and 1 mm, imaged 1 m behind each disc. The disks are illuminated by light of wavelength of 633 nm, diverging from a point 1 m in front of each disc. Each image is 16 mm wide.

Intensity and size
For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. This means that one can compensate for a reduction in the source's wavelength by increasing the distance between the circular object and screen or reducing the circular object's diameter.

The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source (point source at infinity): $$U(P_1, r) \propto J_0^2 \left(\frac{\pi r d}{\lambda b}\right)$$ where
 * r is the distance of the point P1 on the screen from the optical axis
 * d is the diameter of circular object
 * λ is the wavelength
 * b is the distance between circular object and screen.

The following images show the radial intensity distribution of the simulated Arago spot images above:

The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above.

Finite source size and spatial coherence
The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. If the wave source has a finite size S then the Arago spot will have an extent that is given by Sb/g, as if the circular object acted like a lens. At the same time the intensity of the Arago spot is reduced with respect to the intensity of the undisturbed wave front. Defining the relative intensity $$I_\text{rel}$$as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation: $$I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right)$$ where $$J_0$$and $$J_1$$are the Bessel functions of the first kind. $$R$$ is the radius of the disc casting the shadow, $$\lambda$$ the wavelength and $$g$$ the distance between source and disc. For large sources the following asymptotic approximation applies: $$I_\text{rel}(w) \approx \frac{2 g \lambda }{\pi^2 w R}$$

Deviation from circularity
If the cross-section of the circular object deviates slightly from its circular shape (but it still has a sharp edge on a smaller scale) the shape of the point-source Arago spot changes. In particular, if the object has an ellipsoidal cross-section the Arago spot has the shape of an evolute. Note that this is only the case if the source is close to an ideal point source. From an extended source the Arago spot is only affected marginally, since one can interpret the Arago spot as a point-spread function. Therefore, the image of the extended source only becomes washed out due to the convolution with the point-spread function, but it does not decrease in overall intensity.

Surface roughness of circular object
The Arago spot is very sensitive to small-scale deviations from the ideal circular cross-section. This means that a small amount of surface roughness of the circular object can completely cancel out the bright spot. This is shown in the following three diagrams which are simulations of the Arago spot from a 4 mm diameter disc (g = b = 1 m):

The simulation includes a regular sinusoidal corrugation of the circular shape of amplitude 10 μm, 50 μm and 100 μm, respectively. Note, that the 100 μm edge corrugation almost completely removes the central bright spot.

This effect can be best understood using the Fresnel zone concept. The field transmitted by a radial segment that stems from a point on the obstacle edge provides a contribution whose phase is tight to the position of the edge point relative to Fresnel zones. If the variance in the radius of the obstacle are much smaller than the width of Fresnel zone near the edge, the contributions form radial segments are approximately in phase and interfere constructively. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity.

The adjacent Fresnel zone is approximately given by: $$\Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r.$$

The edge corrugation should not be much more than 10% of this width to see a close to ideal Arago spot. In the above simulations with the 4 mm diameter disc the adjacent Fresnel zone has a width of about 77 μm.

Arago spot with matter waves
In 2009, the Arago spot experiment was demonstrated with a supersonic expansion beam of deuterium molecules (an example of neutral matter waves). Material particles behaving like waves is known from quantum mechanics. The wave nature of particles actually dates back to de Broglie's hypothesis as well as Davisson and Germer's experiments. An Arago spot of electrons, which also constitute matter waves, can be observed in transmission electron microscopes when examining circular structures of a certain size.

The observation of an Arago spot with large molecules, thus proving their wave-nature, is a topic of current research.

Other applications
Beside the demonstration of wave-behavior, the Arago spot also has a few other applications. One of the ideas is to use the Arago spot as a straight line reference in alignment systems. Another is to probe aberrations in laser beams by using the spot's sensitivity to beam aberrations. Finally, the aragoscope has been proposed as a method for dramatically improving the diffraction-limited resolution of space-based telescopes.

title: Arago spot


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Battle of Hastings

The Battle of Hastings was fought on 14 October 1066 between the Norman-French army of William, Duke of Normandy, and an English army under the Anglo-Saxon King Harold Godwinson, beginning the Norman Conquest of England. It took place approximately 7 mi northwest of Hastings, close to the present-day town of Battle, East Sussex, and was a decisive Norman victory.

The background to the battle was the death of the childless King Edward the Confessor in January 1066, which set up a succession struggle between several claimants to his throne. Harold was crowned king shortly after Edward's death, but faced invasions by William, his own brother Tostig, and the Norwegian King Harald Hardrada (Harold III of Norway). Hardrada and Tostig defeated a hastily gathered army of Englishmen at the Battle of Fulford on 20 September 1066, and were in turn defeated by Harold at the Battle of Stamford Bridge five days later. The deaths of Tostig and Hardrada at Stamford Bridge left William as Harold's only serious opponent. While Harold and his forces were recovering, William landed his invasion forces in the south of England at Pevensey on 28 September 1066 and established a beachhead for his conquest of the kingdom. Harold was forced to march south swiftly, gathering forces as he went.

The exact numbers present at the battle are unknown as even modern estimates vary considerably. The composition of the forces is clearer: the English army was composed almost entirely of infantry and had few archers, whereas only about half of the invading force was infantry, the rest split equally between cavalry and archers. Harold appears to have tried to surprise William, but scouts found his army and reported its arrival to William, who marched from Hastings to the battlefield to confront Harold. The battle lasted from about 9 am to dusk. Early efforts of the invaders to break the English battle lines had little effect. Therefore, the Normans adopted the tactic of pretending to flee in panic and then turning on their pursuers. Harold's death, probably near the end of the battle, led to the retreat and defeat of most of his army. After further marching and some skirmishes, William was crowned as king on Christmas Day 1066.

There continued to be rebellions and resistance to William's rule, but Hastings effectively marked the culmination of William's conquest of England. Casualty figures are hard to come by, but some historians estimate that 2,000 invaders died along with about twice that number of Englishmen. William founded a monastery at the site of the battle, the high altar of the abbey church supposedly placed at the spot where Harold died.

Background
In 911, the Carolingian ruler Charles the Simple allowed a group of Vikings to settle in Normandy under their leader Rollo. Their settlement proved successful, and they quickly adapted to the indigenous culture, renouncing paganism, converting to Christianity, and intermarrying with the local population. Over time, the frontiers of the duchy expanded to the west. In 1002, King Æthelred II married Emma, the sister of Richard II, Duke of Normandy. Their son Edward the Confessor spent many years in exile in Normandy, and succeeded to the English throne in 1042. This led to the establishment of a powerful Norman interest in English politics, as Edward drew heavily on his former hosts for support, bringing in Norman courtiers, soldiers, and clerics and appointing them to positions of power, particularly in the Church. Edward was childless and embroiled in conflict with the formidable Godwin, Earl of Wessex, and his sons, and he may also have encouraged Duke William of Normandy's ambitions for the English throne.

Succession crisis in England
King Edward's death on 5 January 1066 left no clear heir, and several contenders laid claim to the throne of England. Edward's immediate successor was the Earl of Wessex, Harold Godwinson, the richest and most powerful of the English aristocrats and son of Godwin, Edward's earlier opponent. Harold was elected king by the Witenagemot of England and crowned by Ealdred, the Archbishop of York, although Norman propaganda claimed that the ceremony was performed by Stigand, the uncanonically elected Archbishop of Canterbury. Harold was at once challenged by two powerful neighbouring rulers. Duke William claimed that he had been promised the throne by King Edward and that Harold had sworn agreement to this. Harald Hardrada of Norway also contested the succession. His claim to the throne was based on an agreement between his predecessor Magnus the Good and the earlier King of England Harthacnut, whereby, if either died without heir, the other would inherit both England and Norway. William and Harald Hardrada immediately set about assembling troops and ships for separate invasions.

Tostig and Hardrada's invasions
In early 1066, Harold's exiled brother Tostig Godwinson raided southeastern England with a fleet he had recruited in Flanders, later joined by other ships from Orkney. Threatened by Harold's fleet, Tostig moved north and raided in East Anglia and Lincolnshire. He was driven back to his ships by the brothers Edwin, Earl of Mercia and Morcar, Earl of Northumbria. Deserted by most of his followers, he withdrew to Scotland, where he spent the middle of the year recruiting fresh forces. Hardrada invaded northern England in early September, leading a fleet of more than 300 ships carrying perhaps 15,000 men. Hardrada's army was further augmented by the forces of Tostig, who supported the Norwegian king's bid for the throne. Advancing on York, the Norwegians occupied the city after defeating a northern English army under Edwin and Morcar on 20 September at the Battle of Fulford.

English army and Harold's preparations
The English army was organised along regional lines, with the fyrd, or local levy, serving under a local magnate – whether an earl, bishop, or sheriff. The fyrd was composed of men who owned their own land, and were equipped by their community to fulfil the king's demands for military forces. For every five hides, or units of land nominally capable of supporting one household, one man was supposed to serve. It appears that the hundred was the main organising unit for the fyrd. As a whole, England could furnish about 14,000 men for the fyrd, when it was called out. The fyrd usually served for two months, except in emergencies. It was rare for the whole national fyrd to be called out; between 1046 and 1065 it was only done three times, in 1051, 1052, and 1065. The king also had a group of personal armsmen, known as housecarls, who formed the backbone of the royal forces. Some earls also had their own forces of housecarls. Thegns, the local landowning elites, either fought with the royal housecarls or attached themselves to the forces of an earl or other magnate. The fyrd and the housecarls both fought on foot, with the major difference between them being the housecarls' superior armour. The English army does not appear to have had a significant number of archers.

Harold had spent mid-1066 on the south coast with a large army and fleet waiting for William to invade. The bulk of his forces were militia who needed to harvest their crops, so on 8 September Harold dismissed the militia and the fleet. Learning of the Norwegian invasion he rushed north, gathering forces as he went, and took the Norwegians by surprise, defeating them at the Battle of Stamford Bridge on 25 September. Harald Hardrada and Tostig were killed, and the Norwegians suffered such great losses that only 24 of the original 300 ships were required to carry away the survivors. The English victory came at great cost, as Harold's army was left in a battered and weakened state, and far from the south.

William's preparations and landing
William assembled a large invasion fleet and an army gathered from Normandy and the rest of France, including large contingents from Brittany and Flanders. He spent almost nine months on his preparations, as he had to construct a fleet from nothing. According to some Norman chronicles, he also secured diplomatic support, although the accuracy of the reports has been a matter of historical debate. The most famous claim is that Pope Alexander II gave a papal banner as a token of support, which only appears in William of Poitiers's account, and not in more contemporary narratives. In April 1066 Halley's Comet appeared in the sky, and was widely reported throughout Europe. Contemporary accounts connected the comet's appearance with the succession crisis in England.

William mustered his forces at Saint-Valery-sur-Somme, and was ready to cross the English Channel by about 12 August. But the crossing was delayed, either because of unfavourable weather or to avoid being intercepted by the powerful English fleet. The Normans crossed to England a few days after Harold's victory over the Norwegians, following the dispersal of Harold's naval force, and landed at Pevensey in Sussex on 28 September. A few ships were blown off course and landed at Romney, where the Normans fought the local fyrd. After landing, William's forces built a wooden castle at Hastings, from which they raided the surrounding area. More fortifications were erected at Pevensey.

Norman forces at Hastings

The exact numbers and composition of William's force are unknown. A contemporary document claims that William had 776 ships, but this may be an inflated figure. Figures given by contemporary writers for the size of the army are highly exaggerated, varying from 14,000 to 150,000. Modern historians have offered a range of estimates for the size of William's forces: 7,000–8,000 men including 1,000–2,000 cavalry; 10,000–12,000 men; 10,000 men including 3,000 cavalry; or 7,500 men. The army consisted of about one half infantry, one quarter cavalry, and one quarter archers or crossbowmen. Later lists of companions of William the Conqueror are extant, but most are padded with extra names; only about 35 named individuals can be reliably identified as having been with William at Hastings.

The main armour was chainmail hauberks, usually knee-length, with slits to allow riding, some with sleeves to the elbows. Some hauberks may have been made of scales attached to a tunic, with the scales made of metal, horn or hardened leather. Headgear was usually a conical metal helmet with a vertical band guarding the bridge of the nose. Horsemen and infantry carried shields. The infantryman's shield was usually round and made of wood with metal reinforcement. Horsemen changed to a kite-shaped shield and were usually armed with a lance. The couched lance, carried tucked against the body under the right arm, was relatively new and probably not used at Hastings, as the terrain was unfavourable for long cavalry charges. Both infantry and cavalry usually fought with a straight sword, long and double-edged. The infantry could also use javelins and long spears. Some of the cavalry may have used a mace instead of a sword. Archers would have used a self bow or a crossbow, and most would not have had armour.

Harold moves south
After defeating his brother Tostig and Harald Hardrada in the north, Harold left much of his forces in the north, including Morcar and Edwin, and marched the rest of his army south to deal with the threatened Norman invasion. It is unclear when Harold learned of William's landing, but it was probably during the march. Harold stopped in London for about a week before Hastings, so it is likely that he spent about a week on his march south, averaging about 27 mi per day, for the approximately 200 mi. Harold camped at Caldbec Hill on the night of 13 October, near a "hoar-apple tree", about 8 mi from William's castle at Hastings. Some of the early contemporary French accounts mention an emissary or emissaries sent by Harold to William, which is likely, though nothing came of it.

Although Harold attempted to surprise the Normans, William's scouts reported the English arrival to the duke. The exact events preceding the battle are obscure, with contradictory accounts in the sources, but all agree that William's army advanced from his castle towards the enemy. Harold had taken a defensive position at the top of Senlac Hill (present-day Battle, East Sussex), about 6 mi from William's castle at Hastings.

English forces at Hastings
The exact number of soldiers in Harold's army at Hastings is unknown, as contemporary records do not give reliable figures. Some Norman sources give 400,000 to 1,200,000 on Harold's side, while English sources seem to underestimate Harold's army, perhaps to make the English defeat seem less devastating. Recent historians have suggested figures of between 5,000 and 13,000, while most argue for a figure of 7,000–8,000 English troops. These men would have been a mix of the fyrd and housecarls. Few individual Englishmen are known to have been present; about 20 named individuals can reasonably be assumed to have fought with Harold at Hastings, including Harold's brothers Gyrth and Leofwine and two other relatives.

The English army consisted entirely of infantry. It is possible that some of the higher-class members of the army rode to battle, but then dismounted when battle was joined to fight on foot. The core of the army was made up of housecarls, full-time professional soldiers. Their armour consisted of a conical helmet, a mail hauberk, and a shield, which might be either kite-shaped or round. Most housecarls fought with the two-handed Danish battleaxe, but they could also carry a sword. The rest of the army was made up of levies from the fyrd, also infantry but more lightly armoured and not professionals. Most of the infantry would have formed part of the shield wall, in which the front ranks locked their shields together. Behind them would have been axemen and javelins as well as archers.

Background and location
Because many of the primary accounts contradict each other at times, it is impossible to provide an authoritative description of the battle. The only undisputed facts are that fighting began at 9 am Saturday 14 October 1066 and that the battle lasted until dusk. Sunset on the day of the battle was at 4:54 pm, with the battlefield mostly dark by 5:54 pm and in full darkness by 6:24 pm. Moonrise that night was not until 11:12 pm, so once the sun set, there was little light on the battlefield. William of Jumièges reports that Duke William kept his army armed and ready against a surprise attack for the entire night before. The battle took place 7 mi north of Hastings at the present-day town of Battle, between two hills – Caldbec Hill to the north and Telham Hill to the south. The area was heavily wooded, with a marsh nearby. The name traditionally given to the battle is unusual – there were several settlements much closer to the battlefield than Hastings. The Anglo-Saxon Chronicle called it the battle "at the hoary apple tree". Within 40 years, the battle was described by the Anglo-Norman chronicler Orderic Vitalis as "Senlac", a Norman-French adaptation of the Old English word "Sandlacu", which means "sandy water". This may have been the name of the stream that crosses the battlefield. The battle was already being referred to as "bellum Haestingas" or "Battle of Hastings" by 1086, in the Domesday Book.

The sun rose at 6:48 am that morning, with the day unusually bright. The weather conditions are not recorded. The precise route of the English army southward to the battlefield is not known. Several roads are possible: one, an old Roman road that ran from Rochester to Hastings has long been favoured because of a large coin hoard found nearby in 1876. Another possibility is the Roman road between London and Lewes and then over local tracks to the battlefield. Some accounts of the battle indicate that the Normans advanced from Hastings to the battlefield, but the contemporary account of William of Jumièges places the Normans at the site of the battle the night before. Most historians incline towards the former view,  but M. K. Lawson argues that William of Jumièges's account is correct.

Dispositions of forces and tactics
Harold's forces deployed in a small, dense formation at the top of a steep slope, with their flanks protected by woods and marshy ground in front of them. The line may have extended far enough to be anchored on a nearby stream. The English formed a shield wall, with the front ranks holding their shields close together or even overlapping to provide protection from attack. Sources differ on the exact site that the English fought on: some sources state the site of the abbey,  but some newer sources suggest it was Caldbec Hill.

More is known about the Norman deployment. Duke William appears to have arranged his forces in three groups, or "battles", which roughly corresponded to their origins. The left units were the Bretons, along with those from Anjou, Poitou and Maine. This division was led by Alan the Red, a relative of the Breton count. The centre was held by the Normans, under the direct command of the duke and with many of his relatives and kinsmen grouped around the ducal party. The final division, on the right, consisted of the Frenchmen, along with some men from Picardy, Boulogne, and Flanders. The right was commanded by William fitzOsbern and Count Eustace II of Boulogne. The front lines were made up of archers, with a line of foot soldiers armed with spears behind. There were probably a few crossbowmen and slingers in with the archers. The cavalry was held in reserve, and a small group of clergymen and servants situated at the base of Telham Hill was not expected to take part in the fighting.

William's disposition of his forces implies that he planned to open the battle with archers in the front rank weakening the enemy with arrows, followed by infantry who would engage in close combat. The infantry would create openings in the English lines that could be exploited by a cavalry charge to break through the English forces and pursue the fleeing soldiers.

Beginning of the battle

The battle opened with the Norman archers shooting uphill at the English shield wall, to little effect. The uphill angle meant that the arrows either bounced off the shields of the English or overshot their targets and flew over the top of the hill. The lack of English archers hampered the Norman archers, as there were few English arrows to be gathered up and reused. After the attack from the archers, William sent the spearmen forward to attack the English. They were met with a barrage of missiles, not arrows but spears, axes and stones. The infantry was unable to force openings in the shield wall, and the cavalry advanced in support. The cavalry also failed to make headway, and a general retreat began, blamed on the Breton division on William's left. A rumour started that the duke had been killed, which added to the confusion. The English forces began to pursue the fleeing invaders, but William rode through his forces, showing his face and yelling that he was still alive. The duke then led a counter-attack against the pursuing English forces; some of the English rallied on a hillock before being overwhelmed.

It is not known whether the English pursuit was ordered by Harold or if it was spontaneous. Wace relates that Harold ordered his men to stay in their formations but no other account gives this detail. The Bayeux Tapestry depicts the death of Harold's brothers Gyrth and Leofwine occurring just before the fight around the hillock. This may mean that the two brothers led the pursuit. The Carmen de Hastingae Proelio relates a different story for the death of Gyrth, stating that the duke slew Harold's brother in combat, perhaps thinking that Gyrth was Harold. William of Poitiers states that the bodies of Gyrth and Leofwine were found near Harold's, implying that they died late in the battle. It is possible that if the two brothers died early in the fighting their bodies were taken to Harold, thus accounting for their being found near his body after the battle. The military historian Peter Marren speculates that if Gyrth and Leofwine died early in the battle, that may have influenced Harold to stand and fight to the end.

Feigned flights
A lull probably occurred early in the afternoon, and a break for rest and food would probably have been needed. William may have also needed time to implement a new strategy, which may have been inspired by the English pursuit and subsequent rout by the Normans. If the Normans could send their cavalry against the shield wall and then draw the English into more pursuits, breaks in the English line might form. William of Poitiers says the tactic was used twice. Although arguments have been made that the chroniclers' accounts of this tactic were meant to excuse the flight of the Norman troops from battle, this is unlikely as the earlier flight was not glossed over. It was a tactic used by other Norman armies during the period. Some historians have argued that the story of the use of feigned flight as a deliberate tactic was invented after the battle; however most historians agree that it was used by the Normans at Hastings.

Although the feigned flights did not break the lines, they probably thinned out the housecarls in the English shield wall. The housecarls were replaced with members of the fyrd, and the shield wall held. Archers appear to have been used again before and during an assault by the cavalry and infantry led by the duke. Although 12th-century sources state that the archers were ordered to shoot at a high angle to shoot over the front of the shield wall, there is no trace of such an action in the more contemporary accounts. It is not known how many assaults were launched against the English lines, but some sources record various actions by both Normans and Englishmen that took place during the afternoon's fighting. The Carmen claims that Duke William had two horses killed under him during the fighting, but William of Poitiers's account states that it was three.

Death of Harold

Harold appears to have died late in the battle, although accounts in the various sources are contradictory. William of Poitiers only mentions his death, without giving any details on how it occurred. The Tapestry is not helpful, as it shows a figure holding an arrow sticking out of his eye next to a falling fighter being hit with a sword. Over both figures is a statement "Here King Harold has been killed". It is not clear which figure is meant to be Harold, or if both are meant. The earliest written mention of the traditional account of Harold dying from an arrow to the eye dates to the 1080s from a history of the Normans written by an Italian monk, Amatus of Montecassino. William of Malmesbury stated that Harold died from an arrow to the eye that went into the brain, and that a knight wounded Harold at the same time. Wace repeats the arrow-to-the-eye account. The Carmen states that Duke William killed Harold, but this is unlikely, as such a feat would have been recorded elsewhere. The account of William of Jumièges is even more unlikely, as it has Harold dying in the morning, during the first fighting. The Chronicle of Battle Abbey states that no one knew who killed Harold, as it happened in the press of battle. A modern biographer of Harold, Ian Walker, states that Harold probably died from an arrow in the eye, although he also says it is possible that Harold was struck down by a Norman knight while mortally wounded in the eye. Another biographer of Harold, Peter Rex, after discussing the various accounts, concludes that it is not possible to declare how Harold died.

Harold's death left the English forces leaderless, and they began to collapse. Many of them fled, but the soldiers of the royal household gathered around Harold's body and fought to the end. The Normans began to pursue the fleeing troops, and except for a rearguard action at a site known as the "Malfosse", the battle was over. Exactly what happened at the Malfosse, or "Evil Ditch", and where it took place, is unclear. It occurred at a small fortification or set of trenches where some Englishmen rallied and seriously wounded Eustace of Boulogne before being defeated by the Normans.

Reasons for the outcome
Harold's defeat was probably due to several circumstances. One was the need to defend against two almost simultaneous invasions. The fact that Harold had dismissed his forces in southern England on 8 September also contributed to the defeat. Many historians fault Harold for hurrying south and not gathering more forces before confronting William at Hastings, although it is not clear that the English forces were insufficient to deal with William's forces. Against these arguments for an exhausted English army, the length of the battle, which lasted an entire day, shows that the English forces were not tired by their long march. Tied in with the speed of Harold's advance to Hastings is the possibility Harold may not have trusted Earls Edwin of Mercia and Morcar of Northumbria once their enemy Tostig had been defeated, and declined to bring them and their forces south. Modern historians have pointed out that one reason for Harold's rush to battle was to contain William's depredations and keep him from breaking free of his beachhead.

Most of the blame for the defeat probably lies in the events of the battle. William was the more experienced military leader, and in addition the lack of cavalry on the English side allowed Harold fewer tactical options. Some writers have criticised Harold for not exploiting the opportunity offered by the rumoured death of William early in the battle. The English appear to have erred in not staying strictly on the defensive, for when they pursued the retreating Normans they exposed their flanks to attack. Whether this was due to the inexperience of the English commanders or the indiscipline of the English soldiers is unclear. In the end, Harold's death appears to have been decisive, as it signalled the break-up of the English forces in disarray. The historian David Nicolle said of the battle that William's army "demonstrated – not without difficulty – the superiority of Norman-French mixed cavalry and infantry tactics over the Germanic-Scandinavian infantry traditions of the Anglo-Saxons."

Aftermath
The day after the battle, Harold's body was identified, either by his armour or by marks on his body. His personal standard was presented to William, and later sent to the papacy. The bodies of the English dead, including some of Harold's brothers and housecarls, were left on the battlefield, although some were removed by relatives later. The Norman dead were buried in a large communal grave, which has not been found. Exact casualty figures are unknown. Of the Englishmen known to be at the battle, the number of dead implies that the death rate was about 50 per cent of those engaged, although this may be too high. Of the named Normans who fought at Hastings, one in seven is stated to have died, but these were all noblemen, and it is probable that the death rate among the common soldiers was higher. Although Orderic Vitalis's figures are highly exaggerated, his ratio of one in four casualties may be accurate. Marren speculates that perhaps 2,000 Normans and 4,000 Englishmen were killed at Hastings. Reports stated that some of the English dead were still being found on the hillside years later. Although scholars thought for a long time that remains would not be recoverable, due to the acidic soil, recent finds have changed this view. One skeleton that was found in a medieval cemetery, and originally was thought to be associated with the 13th century Battle of Lewes, now is thought to be associated with Hastings instead.

One story relates that Gytha, Harold's mother, offered the victorious duke the weight of her son's body in gold for its custody, but was refused. William ordered that Harold's body be thrown into the sea, but whether that took place is unclear. Another story relates that Harold was buried at the top of a cliff. Waltham Abbey, which had been founded by Harold, later claimed that his body had been secretly buried there. Other legends claimed that Harold did not die at Hastings, but escaped and became a hermit at Chester.

William expected to receive the submission of the surviving English leaders after his victory, but instead Edgar the Ætheling was proclaimed king by the Witenagemot, with the support of Earls Edwin and Morcar, Stigand, the Archbishop of Canterbury, and Ealdred, the Archbishop of York. William therefore advanced on London, marching around the coast of Kent. He defeated an English force that attacked him at Southwark but was unable to storm London Bridge, forcing him to reach the capital by a more circuitous route.

William moved up the Thames valley to cross the river at Wallingford, where he received the submission of Stigand. He then travelled north-east along the Chilterns, before advancing towards London from the north-west, fighting further engagements against forces from the city. The English leaders surrendered to William at Berkhamsted, Hertfordshire. William was acclaimed King of England and crowned by Ealdred on 25 December 1066, in Westminster Abbey.

Despite the submission of the English nobles, resistance continued for several years. There were rebellions in Exeter in late 1067, an invasion by Harold's sons in mid-1068, and an uprising in Northumbria in 1068. In 1069 William faced more troubles from Northumbrian rebels, an invading Danish fleet, and rebellions in the south and west of England. He ruthlessly put down the various risings, culminating in the Harrying of the North in late 1069 and early 1070 that devastated parts of northern England. A further rebellion in 1070 by Hereward the Wake was also defeated by the king, at Ely.

Battle Abbey was founded by William at the site of the battle. According to 12th-century sources, William made a vow to found the abbey, and the high altar of the church was placed at the site where Harold had died. More likely, the foundation was imposed on William by papal legates in 1070. The topography of the battlefield has been altered by subsequent construction work for the abbey, and the slope defended by the English is now much less steep than it was at the time of the battle; the top of the ridge has also been built up and levelled. After the Dissolution of the Monasteries, the abbey's lands passed to secular landowners, who used it as a residence or country house. In 1976 the estate was put up for sale and purchased by the government with the aid of some American donors who wished to honour the 200th anniversary of American independence. The battlefield and abbey grounds are currently owned and administered by English Heritage and are open to the public.

The Bayeux Tapestry is an embroidered narrative of the events leading up to Hastings probably commissioned by Odo of Bayeux soon after the battle, perhaps to hang at the bishop's palace at Bayeux. In modern times annual reenactments of the Battle of Hastings have drawn thousands of participants and spectators to the site of the original battle.

Some English veterans of the battle left England and joined the Varangian Guard in Constantinople. They fought the Normans again at the Battle of Dyrrhachium in 1081, and were defeated again in similar circumstances.

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

title: Battle of Hastings


text:
Porcupine

Porcupines are large rodents with coats of sharp spines, or quills, that protect them against predation. The term covers two families of animals: the Old World porcupines of the family Hystricidae, and the New World porcupines of the family Erethizontidae. Both families belong to the infraorder Hystricognathi within the profoundly diverse order Rodentia and display superficially similar coats of rigid or semi-rigid quills, which are modified hairs composed of keratin. Despite this, the two groups are distinct from one another and are not closely related to each other within the Hystricognathi. The largest species of porcupine is the third-largest living rodent in the world, after the capybara and beaver.

The Old World porcupines (Hystricidae) live in Italy, Asia (western and southern), and most of Africa. They are large, terrestrial, and strictly nocturnal.

The New World porcupines (Erethizontidae) are indigenous to North America and northern South America. They live in wooded areas and can climb trees, where some species spend their entire lives. They are less strictly nocturnal than their Old World counterparts and generally smaller.

Most porcupines are about 25 - 36 in long, with a 8 - 10 in long tail. Weighing 12 - 35 lb, they are rounded, large, and slow, and use an aposematic strategy of defence. Porcupines' colouration consists of various shades of brown, grey and white. Porcupines' spiny protection resembles that of the only distantly related erinaceomorph hedgehogs and Australian monotreme echidnas as well as tenrecid tenrecs.

Etymology
The word "porcupine" comes from the Latin porcus pig + spina spine, quill, via Italian (Italian "porcospino", thorn-pig)—Middle French—Middle English. A regional American name for the animal is "quill-pig".

A baby porcupine is a porcupette. When born, a porcupette's quills are soft hair; they harden within a few days, forming the sharp quills of adults.

Evolution
Fossils belonging to the genus Hystrix date back to the late Miocene of the continent of Africa.

Taxonomy
A porcupine is any of 30 species of rodents belonging to the families Erethizontidae (genera: Coendou, Erethizon, and Chaetomys) or Hystricidae (genera: Atherurus, Hystrix, and Trichys). Porcupines vary in size considerably: Rothschild's porcupine of South America weighs less than a kilogram (2.2 lb); the crested porcupine found in Italy, North Africa, and sub-Saharan Africa can grow to well over 27 kg. The two families of porcupines are quite different, and although both belong to the Hystricognathi branch of the vast order Rodentia, they are not closely related.

Old World compared with New World species
The 11 Old World porcupines tend to be fairly large and have spines grouped in clusters.

The two subfamilies of New World porcupines are mostly smaller (although the North American porcupine reaches about 85 cm in length and 18 kg), have their quills attached singly rather than grouped in clusters, and are excellent climbers, spending much of their time in trees. The New World porcupines evolved their spines independently (through convergent evolution) and are more closely related to several other families of rodents than they are to the Old World porcupines.

Longevity
Porcupines have a relatively high longevity and hold the record for being the longest-living rodent, with one individual named Cooper living over 32 years.

Diet
The North American porcupine is a herbivore and often climbs trees for food; it eats leaves, herbs, twigs, and green plants such as clover. In the winter, it may eat bark. The African porcupine is not a climber; instead, it forages on the ground. It is mostly nocturnal but will sometimes forage for food during the day, eating bark, roots, fruits, berries, and farm crops. Porcupines have become a pest in Kenya and are eaten as a delicacy.

Defence
Defensive behaviour displays in a porcupine depend on sight, scent, and sound. Often, these displays are shown when a porcupine becomes agitated or annoyed. There are four main displays seen in a porcupine: (in order from least to most aggressive) quill erection, teeth clattering, odour emission, and attack. A porcupine's colouring aids in part of its defence as most of the predators are nocturnal and colour-blind. A porcupine's markings are black and white. The dark body and coarse hair of the porcupine are dark brown/black and when quills are raised, present a white strip down its back mimicking the look of a skunk. This, along with the raising of the sharp quills, deters predators. Along with the raising of the quills, porcupines clatter their teeth to warn predators not to approach. The incisors vibrate against each other, the strike zone shifts back, and the cheek teeth clatter. This behaviour is often paired with body shivering, which is used to further display the dangerous quills. The rattling of quills is aided by the hollow quills at the back end of the porcupine. The use of odour is when the sight and sound have failed. An unpleasant scent is produced from the skin above the tail in times of stress and is often seen with a quill erection. If these processes fail, the porcupine will attack by running sideways or backwards into predators. A porcupine's tail can also be swung in the direction of the predator; if contact is made, the quills could be impaled into the predator causing injury or death.

Quills
Porcupines' quills, or spines, take on various forms, depending on the species, but all are modified hairs coated with thick plates of keratin, and embedded in the skin musculature. Old World porcupines have quills embedded in clusters, whereas in New World porcupines, single quills are interspersed with bristles, underfur, and hair.

Quills are released by contact or may drop out when the porcupine shakes its body. New quills grow to replace lost ones. Despite what is commonly believed, porcupines can't launch their quills at range.

There are some possible antibiotic properties within the quills, specifically associated with the free fatty acids coating the quills. The antibiotic properties are believed to aid a porcupine that has suffered from self-injury.

Uses by humans
Porcupines are seldom eaten in Western culture but are eaten often in Southeast Asia, particularly Vietnam, where the prominent use of them as a food source has contributed to declines in porcupine populations.

Naturalist William J. Long reported the taste of the North American porcupine as "vile" and "malodorous" and delightful only to a lover of strong cheese. With regards to a Maine state law that restricted the killing of porcupines to keep them available as an emergency game for people lost in the woods, he noted: "It is undoubtedly a good law; but I cannot now imagine anyone being grateful for it, unless the stern alternative were death or porcupine."

More commonly, their quills and guard hairs are used for traditional decorative clothing; for example, their guard hairs are used in the creation of the Native American "porky roach" headdress. The main quills may be dyed and then applied in combination with thread to embellish leather accessories, such as knife sheaths and leather bags. Lakota women would harvest the quills for quillwork by throwing a blanket over a porcupine and retrieving the quills left stuck in the blanket.

The presence of barbs, acting like anchors, causes increased pain when removing a quill that has pierced the skin. The shape of the barbs makes the quills effective for penetrating the skin and for remaining in place. The quills have inspired research for such applications as the design of hypodermic needles and surgical staples. In contrast to the current design for surgical staples, the porcupine quill and barb design would allow easy and painless insertion, as the staple would stay in the skin using the anchored barb design rather than being bent under the skin like traditional staples. Porcupines are also sometimes kept as exotic pets.

Habitat
Porcupines occupy a small range of habitats in tropical and temperate parts of Asia, Southern Europe, Africa, and North and South America. They live in forests and deserts, rocky outcrops, and hillsides. Some New World porcupines live in trees, but Old World porcupines prefer a rocky environment. Porcupines can be found on rocky areas up to 3700 m high. They are generally nocturnal but are occasionally active during daylight.

Classification

Porcupines are distributed into two evolutionarily independent groups within the suborder Hystricomorpha of the Rodentia.

 * Infraorder Hystricognathi
 * Family Hystricidae: Old World porcupines
 * African brush-tailed porcupine, Atherurus africanus
 * African crested porcupine, Hystrix cristata
 * Asiatic brush-tailed porcupine, Atherurus macrourus
 * Cape porcupine, Hystrix africaeaustralis
 * Indian porcupine, Hystrix indicus
 * Malayan porcupine, Hystrix brachyura
 * Himalayan porcupine, Hystrix (brachyura) hodgsoni
 * Sunda porcupine, Hystrix javanica
 * Sumatran porcupine, Hystrix (Thecurus) sumatrae
 * Thick-spined porcupine, Hystrix (Thecurus) crassispinis
 * Philippine porcupine, Hystrix (Thecurus) pumilis
 * Long-tailed porcupine, Trichys fasciculata
 * Parvorder Phiomorpha sensu stricto
 * Family Thryonomyidae: cane rats
 * Family Petromuridae: Dassie rats
 * Family Bathyergidae: African mole-rats
 * Parvorder Caviomorpha
 * Superfamily Erethizontoidea
 * Family Erethizontidae: New World porcupines
 * North American porcupine, Erethizon dorsatum
 * Brazilian porcupine, Coendou prehensilis
 * Bicolored-spined porcupine, Coendou bicolor
 * Andean porcupine, Coendou quichua
 * Black dwarf (Koopman's) porcupine, Coendou nycthemera (koopmani)
 * Rothschild's porcupine, Coendou rothschildi
 * Santa Marta porcupine, Coendou sanctemartae
 * Mexican hairy dwarf porcupine, Coendou mexicanus
 * Paraguaian hairy dwarf porcupine, Coendou spinosus
 * Bahia porcupine, Coendou insidiosus
 * Brown hairy dwarf porcupine, Coendou vestitus
 * Streaked dwarf porcupine, Coendou ichillus
 * Black-tailed hairy dwarf porcupine, Coendou melanurus
 * Roosmalen's dwarf porcupine, Coendou roosmalenorum
 * Frosted hairy dwarf porcupine, Coendou pruinosus
 * Stump-tailed porcupine, Coendou rufescens
 * Bristle-spined porcupine, Chaetomys subspinosus (sometimes considered an echimyid)
 * Superfamily Cavioidea
 * Family Hydrochaeridae: capybara
 * Family Caviidae: Guinea-pigs
 * Family Dasyproctidae: agoutis and acouchis
 * Superfamily Octodontoidea
 * Family Abrocomidae: chinchilla-rats
 * Family Octodontidae: degus
 * Family Ctenomyidae: tuco-tucos
 * Family Echimyidae: spiny rats
 * Family Myocastoridae: nutrias
 * Family Capromyidae: hutias
 * Superfamily Chinchilloidea
 * Family Chinchillidae: chinchillas and allies
 * Family Dinomyidae: pacaranas

title: Porcupine


text:
François Arago

 Dominique François Jean Arago (Domènec Francesc Joan Aragó), known simply as François Arago (Catalan: Francesc Aragó, ; 26 February 1786 – 2 October 1853), was a French mathematician, physicist, astronomer, freemason, supporter of the Carbonari revolutionaries and politician.

Early life and work
Arago was born at Estagel, a small village of 3,000 near Perpignan, in the département of Pyrénées-Orientales, France, where his father held the position of Treasurer of the Mint. His parents were François Bonaventure Arago (1754–1814) and Marie Arago (1755–1845).

Arago was the eldest of four brothers. Jean (1788–1836) emigrated to North America and became a general in the Mexican army. Jacques Étienne Victor (1799–1855) took part in Louis de Freycinet's exploring voyage in the Uranie from 1817 to 1821, and on his return to France devoted himself to his journalism and the drama. The fourth brother, Étienne Vincent (1802–1892), is said to have collaborated with Honoré de Balzac in The Heiress of Birague, and from 1822 to 1847 wrote a great number of light dramatic pieces, mostly in collaboration.

Showing decided military tastes, François Arago was sent to the municipal college of Perpignan, where he began to study mathematics in preparation for the entrance examination of the École Polytechnique. Within two years and a half he had mastered all the subjects prescribed for examination, and a great deal more, and, on going up for examination at Toulouse, he astounded his examiner by his knowledge of J.-L. Lagrange's work.

Towards the close of 1803, Arago entered the École Polytechnique, Paris, but apparently found the professors there incapable of imparting knowledge or maintaining discipline. The artillery service was his ambition, and in 1804, through the advice and recommendation of Siméon Poisson, he received the appointment of secretary to the Paris Observatory. He now became acquainted with Pierre-Simon Laplace, and through his influence was commissioned, with Jean-Baptiste Biot, to complete the meridian arc measurements which had been begun by J. B. J. Delambre, and interrupted since the death of P. F. A. Méchain in 1804 (the meridian arc of Delambre and Méchain). Arago and Biot left Paris in 1806 and began operations along the mountains of Spain. Biot returned to Paris after they had determined the latitude of Formentera, the southernmost point to which they were to carry the survey. Arago continued the work until 1809, his purpose being to measure a meridian arc in order to determine the exact length of a metre.

After Biot's departure, the political ferment caused by the entrance of the French into Spain extended to the Balearic Islands, and the population suspected Arago's movements and his lighting of fires on the top of Mount Galatzó (Catalan: Mola de l'Esclop) as the activities of a spy for the invading army. Their reaction was such that he was obliged to give himself up for imprisonment in the fortress of Bellver in June 1808. On 28 July he escaped from the island in a fishing-boat, and after an adventurous voyage he reached Algiers on 3 August. From there he obtained a passage in a vessel bound for Marseille, but on 16 August, just as the vessel was nearing Marseille, it fell into the hands of a Spanish corsair. With the rest the crew, Arago was taken to Roses, and imprisoned first in a windmill, and afterwards in a fortress, until the town fell into the hands of the French, when the prisoners were transferred to Palamos.

After three months' imprisonment, Arago and the others were released on the demand of the dey of Algiers, and again set sail for Marseille on 28 November, but then within sight of their port they were driven back by a northerly wind to Bougie on the coast of Africa. Transport to Algiers by sea from this place would have occasioned a weary delay of three months; Arago, therefore, set out over land, guided by a Muslim priest, and reached it on Christmas Day. After six months in Algiers he once again, on 21 June 1809, set sail for Marseille, where he had to undergo a monotonous and inhospitable quarantine in the lazaretto, before his difficulties were over. The first letter he received, while in the lazaretto, was from Alexander von Humboldt; and this was the origin of a connection which, in Arago's words, "lasted over forty years without a single cloud ever having troubled it."

Scientific studies
Arago had succeeded in preserving the records of his survey; and his first act on his return home was to deposit them in the Bureau des Longitudes at Paris. As a reward for his adventurous conduct in the cause of science, he was elected a member of the French Academy of Sciences, at the remarkably early age of twenty-three, and before the close of 1809 he was chosen by the council of the École Polytechnique to succeed Gaspard Monge in the chair of analytical geometry. At the same time he was named by the emperor one of the astronomers of the Paris Observatory, which was accordingly his residence till his death. It was in this capacity that he delivered his remarkably successful series of popular lectures in astronomy, which were continued from 1812 to 1845.

In 1818 or 1819 he proceeded along with Biot to execute geodetic operations on the coasts of France, England and Scotland. They measured the length of the seconds pendulum at Leith, Scotland, and in the Shetland Islands, the results of the observations being published in 1821, along with those made in Spain. Arago was elected a member of the Bureau des Longitudes immediately afterwards, and contributed to each of its Annuals, for about twenty-two years, important scientific notices on astronomy and meteorology and occasionally on civil engineering, as well as interesting memoirs of members of the Academy.

Arago's earliest physical researches were on the pressure of steam at different temperatures, and the velocity of sound, 1818 to 1822. His magnetic observations mostly took place from 1823 to 1826. He discovered rotatory magnetism, what has been called Arago's rotations, and the fact that most bodies could be magnetized; these discoveries were completed and explained by Michael Faraday.

Arago warmly supported Augustin-Jean Fresnel's optical theories, helping to confirm Fresnel's wave theory of light by observing what is now known as the spot of Arago. The two philosophers conducted together those experiments on the polarization of light which led to the inference that the vibrations of the luminiferous ether were transverse to the direction of motion, and that polarization consisted of a resolution of rectilinear propagation into components at right angles to each other. The subsequent invention of the polariscope and discovery of Rotary polarization are due to Arago. He invented the first polarization filter in 1812. He was the first to perform a polarimetric observation of a comet when he discovered polarized light from the tail of the Great Comet of 1819.

The general idea of the experimental determination of the velocity of light in the manner subsequently effected by Hippolyte Fizeau and Léon Foucault was suggested by Arago in 1838, but his failing eyesight prevented his arranging the details or making the experiments.

Arago's fame as an experimenter and discoverer rests mainly on his contributions to magnetism in the co-discovery with Léon Foucault of eddy currents, and still more to optics. He showed that a magnetic needle, made to oscillate over nonferrous surfaces, such as water, glass, copper, etc., falls more rapidly in the extent of its oscillations according as it is more or less approached to the surface. This discovery, which earned him the Copley Medal of the Royal Society in 1825, was followed by another, that a rotating plate of copper tends to communicate its motion to a magnetic needle suspended over it, which he called "magnetism of rotation" but (after Faraday's explanation of 1832 ) is now known as eddy current. Arago is also fairly entitled to be regarded as having proved the long-suspected connexion between the aurora borealis and the variations of the magnetic elements. In 1827 he was elected an associated member of the Royal Institute of the Netherlands, when that institute became the Royal Netherlands Academy of Arts and Sciences in 1851, he became foreign member. In 1828, he was elected a foreign member of the Royal Swedish Academy of Sciences.

In optics, Arago not only made important optical discoveries on his own, but is credited with stimulating the genius of Jean-Augustin Fresnel, with whose history, as well as that of Étienne-Louis Malus and Thomas Young, this part of his life is closely interwoven.

Shortly after the beginning of the 19th century the labours of at least three philosophers were shaping the doctrine of the undulatory, or wave, theory of light. Fresnel's arguments in favour of that theory found little favour with Laplace, Poisson and Biot, the champions of the emission theory; but they were ardently espoused by Humboldt and by Arago, who had been appointed by the Academy to report on the paper. This was the foundation of an intimate friendship between Arago and Fresnel, and of a determination to carry on together further fundamental laws of the polarization of light known by their means. As a result of this work, Arago constructed a polariscope, which he used for some interesting observations on the polarization of the light of the sky. He also discovered the power of rotatory polarization exhibited by quartz.

Among Arago's many contributions to the support of the undulatory hypothesis, comes the experimentum crucis which he proposed to carry out for measuring directly the velocity of light in air and in water and glass. On the emission theory the velocity should be accelerated by an increase of density in the medium; on the wave theory, it should be retarded. In 1838 he communicated to the Academy the details of his apparatus, which utilized the relaying mirrors employed by Charles Wheatstone in 1835 for measuring the velocity of the electric discharge; but owing to the great care required in the carrying out of the project, and to the interruption to his labours caused by the revolution of 1848, it was the spring of 1850 before he was ready to put his idea to the test; and then his eyesight suddenly gave way. Before his death, however, the retardation of light in denser media was demonstrated by the experiments of H. L. Fizeau and B. L. Foucault, which, with improvements in detail, were based on the plan proposed by him.

Politics and legacy

In 1830, Arago, who always professed liberal opinions of the republican type, was elected a member of the chamber of deputies for the Pyrénées-Orientales département, and he employed his talents of eloquence and scientific knowledge in all questions connected with public education, the rewards of inventors, and the encouragement of the mechanical and practical sciences. Many of the most creditable national enterprises, dating from this period, are due to his advocacy – such as the reward to Louis Daguerre for the invention of photography, the grant for the publication of the works of Fermat and Laplace, the acquisition of the museum of Cluny, the development of railways and electric telegraphs, the improvement of the reneile. In 1839, Arago reported the invention of photography to stunned listeners of a joint meeting of the academies of Arts and Sciences.

In 1830, Arago also was appointed director of the Observatory, and as a member of the chamber of deputies he was able to obtain grants of money for rebuilding it in part, and for the addition of magnificent instruments. In the same year, too, he was chosen perpetual secretary of the Academy of Sciences, the place of Joseph Fourier. Arago threw himself into its service, and by his faculty of making friends he gained at once for it and for himself a worldwide reputation. As perpetual secretary it was his duty to pronounce historical eulogies on deceased members; and for this duty his rapidity and facility of thought, and his happy piquancy of style, and his extensive knowledge peculiarly adapted him. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1832.

In 1834, Arago again visited Scotland, to attend the meeting of the British Association at Edinburgh. From this time till 1848 he led a life of comparative quiet – although he continued to work within the Academy and the Observatory to produce a multitude of contributions to all departments of physical science – but on the fall of Louis-Philippe he left his laboratory to join the Provisional Government (24 February 1848). He was entrusted with two important functions, that had never before been given to one person, viz. the ministry of marine and colonies (24 February 1848 – 11 May 1848) and ministry of war (5 April 1848 – 11 May 1848); in the former capacity he improved rations in the navy and abolished flogging. He also abolished political oaths of all kinds and, against an array of moneyed interests, succeeded in procuring the abolition of slavery in the French colonies.

On 10 May 1848, Arago was elected a member of the Executive Power Commission, a governing body of the French Republic. He was made President of the Executive Power Commission (11 May 1848) and served in this capacity as provisional head of state until 24 June 1848, when collective resignation of the commission was submitted to the National Constituent Assembly. At the beginning of May 1852, when the government of Louis Napoleon required an oath of allegiance from all its functionaries, Arago peremptorily refused, and sent in his resignation of his post as astronomer at the Bureau des Longitudes. This, however, the prince president declined to accept, and made "an exception in favour of a savant whose works had thrown lustre on France, and whose existence the government would regret to embitter."

Cape Gregory in Oregon was named by Captain Cook on 12 March 1778 after Saint Gregory, the saint of that day; it was renamed Cape Arago after François Arago.

Last years
Arago remained a consistent republican to the end, and after the coup d'état of 1852, though suffering first from diabetes, then from Bright's disease, complicated by dropsy, he resigned his post as astronomer rather than take the oath of allegiance. Napoleon III gave directions that the old man should be in no way disturbed, and should be left free to say and do what he liked. In the summer of 1853 Arago was advised by his physicians to try the effect of his native air, and he accordingly set out to the eastern Pyrenees, but this was ineffective and he died in Paris. His grave is at the famous Père Lachaise Cemetery in Paris. Arago was an atheist.

Named after Arago

 * The study association for Applied Physics at the University of Twente was named after Arago.
 * His name is one of the 72 names inscribed on the Eiffel Tower.
 * The outer main-belt asteroid 1005 Arago, an inner ring of Neptune, the lunar crater Arago as well as the Martian crater Arago were also named in his honor.
 * Two French cable ships were named after him, the François Arago of 1882 and the Arago of 1914/1931.

Honours

 * Kingdom of Belgium: Officer of the Order of Leopold.
 * 1842 Prussian Pour le Mérite for Sciences and Arts.

Publications
Arago's works were published after his death under the direction J. A. Barral, in 17 vols., 8vo, 1854–1862 ; also separately his Astronomie populaire, in 4 vols.; Notices biographiques, in 3 vols.; Indices scientifiques, in 5 vols.; Voyages scientifiques, in 1 vol.; Grimoires scientifiques, in 2 vols.; Mélanges, in I vol.; and Tables analytiques et documents importants (with portrait), in 1 vol.

English translations of the following portions of Arago's works have appeared:
 * Treatise on Comets, by C. Gold, C.B. (London, 1833); also translated W. H. Smyth and Grant (London, 1861)
 * Euloge of James Watt, by Muirhead (London, 1839); also translated, with notes, by Brougham
 * Popular Lectures on Astronomy, by Walter Kelly and Rev. L. Tomlinson (London, 1854); also translated by Dr W. H. Smyth and Prof. R. Grant, 2 vols. (London, 1855)
 * Arago's Autography, translated by the Rev. Baden Powell (London, 1855, 58)
 * Arago's Meteorological Essays, with introduction by Alexander von Humboldt, translated under the supervision of Colonel Edward Sabine (London, 1855)

title: François Arago


text:
Siméon Denis Poisson

Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Arago spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel.

Biography
Poisson was born in Pithiviers, Loiret district in France, the son of Siméon Poisson, an officer in the French army.

In 1798, he entered the École Polytechnique in Paris as first in his year, and immediately began to attract the notice of the professors of the school, who left him free to make his own decisions as to what he would study. In his final year of study, less than two years after his entry, he published two memoirs, one on Étienne Bézout's method of elimination, the other on the number of integrals of a finite difference equation and this was so impressive that he was allowed to graduate in 1800 without taking the final examination ,. The latter of the memoirs was examined by Sylvestre-François Lacroix and Adrien-Marie Legendre, who recommended that it should be published in the Recueil des savants étrangers, an unprecedented honor for a youth of eighteen. This success at once procured entry for Poisson into scientific circles. Joseph Louis Lagrange, whose lectures on the theory of functions he attended at the École Polytechnique, recognized his talent early on, and became his friend. Meanwhile, Pierre-Simon Laplace, in whose footsteps Poisson followed, regarded him almost as his son. The rest of his career, until his death in Sceaux near Paris, was occupied by the composition and publication of his many works and in fulfilling the duties of the numerous educational positions to which he was successively appointed.

Immediately after finishing his studies at the École Polytechnique, he was appointed répétiteur (teaching assistant) there, a position which he had occupied as an amateur while still a pupil in the school; for his schoolmates had made a custom of visiting him in his room after an unusually difficult lecture to hear him repeat and explain it. He was made deputy professor (professeur suppléant) in 1802, and, in 1806 full professor succeeding Jean Baptiste Joseph Fourier, whom Napoleon had sent to Grenoble. In 1808 he became astronomer to the Bureau des Longitudes; and when the Faculté des sciences de Paris was instituted in 1809 he was appointed a professor of rational mechanics (professeur de mécanique rationelle). He went on to become a member of the Institute in 1812, examiner at the military school (École Militaire) at Saint-Cyr in 1815, graduation examiner at the École Polytechnique in 1816, councillor of the university in 1820, and geometer to the Bureau des Longitudes succeeding Pierre-Simon Laplace in 1827.

In 1817, he married Nancy de Bardi and with her, he had four children. His father, whose early experiences had led him to hate aristocrats, bred him in the stern creed of the First Republic. Throughout the Revolution, the Empire, and the following restoration, Poisson was not interested in politics, concentrating instead on mathematics. He was appointed to the dignity of baron in 1825, but he neither took out the diploma nor used the title. In March 1818, he was elected a Fellow of the Royal Society, in 1822 a Foreign Honorary Member of the American Academy of Arts and Sciences, and in 1823 a foreign member of the Royal Swedish Academy of Sciences. The revolution of July 1830 threatened him with the loss of all his honours; but this disgrace to the government of Louis-Philippe was adroitly averted by François Jean Dominique Arago, who, while his "revocation" was being plotted by the council of ministers, procured him an invitation to dine at the Palais-Royal, where he was openly and effusively received by the citizen king, who "remembered" him. After this, of course, his degradation was impossible, and seven years later he was made a peer of France, not for political reasons, but as a representative of French science. As a teacher of mathematics Poisson is said to have been extraordinarily successful, as might have been expected from his early promise as a répétiteur at the École Polytechnique. Notwithstanding his many official duties, he found time to publish more than three hundred works, several of them extensive treatises, and many of them memoirs dealing with the most abstruse branches of pure mathematics, applied mathematics, mathematical physics, and rational mechanics. (Arago attributed to him the quote, "Life is good for only two things: doing mathematics and teaching it." )

A list of Poisson's works, drawn up by himself, is given at the end of Arago's biography. All that is possible is a brief mention of the more important ones. It was in the application of mathematics to physics that his greatest services to science were performed. Perhaps the most original, and certainly the most permanent in their influence, were his memoirs on the theory of electricity and magnetism, which virtually created a new branch of mathematical physics.

Next (or in the opinion of some, first) in importance stand the memoirs on celestial mechanics, in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important of these are his memoirs Sur les inégalités séculaires des moyens mouvements des planètes, Sur la variation des constantes arbitraires dans les questions de mécanique, both published in the Journal of the École Polytechnique (1809); Sur la libration de la lune, in Connaissance des temps (1821), etc.; and Sur le mouvement de la terre autour de son centre de gravité, in Mémoires de l'Académie (1827), etc. In the first of these memoirs, Poisson discusses the famous question of the stability of the planetary orbits, which had already been settled by Lagrange to the first degree of approximation for the disturbing forces. Poisson showed that the result could be extended to a second approximation, and thus made an important advance in planetary theory. The memoir is remarkable inasmuch as it roused Lagrange, after an interval of inactivity, to compose in his old age one of the greatest of his memoirs, entitled Sur la théorie des variations des éléments des planètes, et en particulier des variations des grands axes de leurs orbites. So highly did he think of Poisson's memoir that he made a copy of it with his own hand, which was found among his papers after his death. Poisson made important contributions to the theory of attraction.

As a tribute to Poisson's scientific work, which stretched to more than 300 publications, he was awarded a French peerage in 1837.

His is one of the 72 names inscribed on the Eiffel Tower.

Poisson's equation
In the theory of potentials, Poisson's equation,

 * $$ \nabla^2 \phi = - 4 \pi \rho, \; $$

is a well-known generalization of Laplace's equation of the second order partial differential equation $$ \nabla^2 \phi = 0$$ for potential $$\phi$$.

If $$ \rho(x, y, z) $$ is a continuous function and if for $$ r \rightarrow \infty $$ (or if a point 'moves' to infinity) a function $$ \phi $$ goes to 0 fast enough, the solution of Poisson's equation is the Newtonian potential

 * $$ \phi = - {1\over 4 \pi} \iiint \frac{\rho (x, y, z)}{ r} \, dV, \; $$

where $$ r $$ is a distance between a volume element $$ dV $$and a point $$ P $$. The integration runs over the whole space.

Poisson's equation was first published in the Bulletin de la société philomatique (1813). Poisson's two most important memoirs on the subject are Sur l'attraction des sphéroides (Connaiss. ft. temps, 1829), and Sur l'attraction d'un ellipsoide homogène (Mim. ft. l'acad., 1835).

Poisson discovered that Laplace's equation is valid only outside of a solid. A rigorous proof for masses with variable density was first given by Carl Friedrich Gauss in 1839. Poisson's equation is applicable in not just gravitation, but also electricity and magnetism.

Electricity and magnetism
As the eighteenth century came to a close, human understanding of electrostatics approached maturity. Benjamin Franklin had already established the notion of electric charge and the conservation of charge; Charles-Augustin de Coulomb had enunciated his inverse-square law of electrostatics. In 1777, Joseph-Louis Lagrange introduced the concept of a potential function that can be used to compute the gravitational force of an extended body. In 1812, Poisson adopted this idea and obtained the appropriate expression for electricity, which relates the potential function $$V$$ to the electric charge density $$\rho$$. Poisson's work on potential theory inspired George Green's 1828 paper, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.

In 1820, Hans Christian Ørsted demonstrated that it was possible to deflect a magnetic needle by closing or opening an electric circuit nearby, resulting in a deluge of published papers attempting to explain the phenomenon. Ampère's law and the Biot-Savart law were quickly deduced. The science of electromagnetism was born. Poisson was also investigating the phenomenon of magnetism at this time, though he insisted on treating electricity and magnetism as separate phenomena. He published two memoirs on magnetism in 1826. By the 1830s, a major research question in the study of electricity was whether or not electricity was a fluid or fluids distinct from matter, or something that simply acts on matter like gravity. Coulomb, Ampère, and Poisson thought that electricity was a fluid distinct from matter. In his experimental research, starting with electrolysis, Michael Faraday sought to show this was not the case. Electricity, Faraday believed, was a part of matter.

Optics
Poisson was a member of the academic "old guard" at the Académie royale des sciences de l'Institut de France, who were staunch believers in the particle theory of light and were skeptical of its alternative, the wave theory. In 1818, the Académie set the topic of their prize as diffraction. One of the participants, civil engineer and opticist Augustin-Jean Fresnel submitted a thesis explaining diffraction derived from analysis of both the Huygens–Fresnel principle and Young's double slit experiment.

Poisson studied Fresnel's theory in detail and looked for a way to prove it wrong. Poisson thought that he had found a flaw when he demonstrated that Fresnel's theory predicts an on-axis bright spot in the shadow of a circular obstacle blocking a point source of light, where the particle-theory of light predicts complete darkness. Poisson argued this was absurd and Fresnel's model was wrong. (Such a spot is not easily observed in everyday situations, because most everyday sources of light are not good point sources.)

The head of the committee, Dominique-François-Jean Arago, performed the experiment. He molded a 2 mm metallic disk to a glass plate with wax. To everyone's surprise he observed the predicted bright spot, which vindicated the wave model. Fresnel won the competition.

After that, the corpuscular theory of light was dead, but was revived in the twentieth century in a different form, wave-particle duality. Arago later noted that the diffraction bright spot (which later became known as both the Arago spot and the Poisson spot) had already been observed by Joseph-Nicolas Delisle and Giacomo F. Maraldi a century earlier.

Pure mathematics and statistics
In pure mathematics, Poisson's most important works were his series of memoirs on definite integrals and his discussion of Fourier series, the latter paving the way for the classic researches of Peter Gustav Lejeune Dirichlet and Bernhard Riemann on the same subject; these are to be found in the Journal of the École Polytechnique from 1813 to 1823, and in the Memoirs de l'Académie for 1823. He also studied Fourier integrals.

Poisson wrote an essay on the calculus of variations (Mem. de l'acad., 1833), and memoirs on the probability of the mean results of observations (Connaiss. d. temps, 1827, &c). The Poisson distribution in probability theory is named after him.

In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so.

In 1829, Poisson published a paper on elastic bodies that contained a statement and proof of a special case of what became known as the divergence theorem.

Analytical mechanics and the calculus of variations
Founded mainly by Leonhard Euler and Joseph-Louis Lagrange in the eighteenth century, the calculus of variations saw further development and applications in the nineteenth.

Let"$S = \int\limits_{a}^{b} f (x, y(x), y'(x)) \, dx,$"where $$y' = \frac{dy}{dx}$$. Then $$S$$ is extremized if it satisfies the Euler–Lagrange equations"$\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0.$"But if $$S$$ depends on higher-order derivatives of $$y(x)$$, that is, if "$S = \int\limits_{a}^{b} f \left(x, y(x), y'(x), ..., y^{(n)}(x) \right) \, dx,$"then $$y$$ must satisfy the Euler–Poisson equation,"$\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) + ... + (-1)^{n} \frac{d^n}{dx^n} \left[ \frac{\partial f}{\partial y^{(n)}} \right]= 0.$ |undefined"Poisson's Traité de mécanique (2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work.

Let $$ q$$ be the position, $$T$$ be the kinetic energy, $$V$$ the potential energy, both independent of time $$t$$. Lagrange's equation of motion reads "$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} = 0, ~ i = 1, 2, ..., n.$"Here, the dot notation for the time derivative is used, $$\frac{dq}{dt} = \dot{q}$$. Poisson set $$L = T - V$$. He argued that if $$V$$ is independent of $$\dot{q}_i$$, he could write"$\frac{\partial L}{\partial \dot{q}_i} = \frac{\partial T}{\partial \dot{q}_i},$"giving "$\frac{d}{dt} \left (\frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0.$"He introduced an explicit formula for momenta, "$ p_i = \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial T}{\partial \dot{q}_i}.$"Thus, from the equation of motion, he got "$ \dot{p}_i = \frac{\partial L}{\partial q_i}.$"Poisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842. Let $$u$$ and $$v$$ be functions of the canonical variables of motion $$q$$ and $$p$$. Then their Poisson bracket is given by"$[u, v] = \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} - \frac{\partial u}{\partial p_i} \frac{\partial v}{\partial q_i}.$"Evidently, the operation anti-commutes. More precisely, $$[u, v] = -[v, u]$$. By Hamilton's equations of motion, the total time derivative of $$u = u (q, p, t)$$ is $$\begin{align} \frac{du}{dt} &= \frac{\partial u}{\partial q_i} \dot{q}_i + \frac{\partial u}{\partial p_i} \dot{p}_i + \frac{\partial u}{\partial t} \\[6pt] &= \frac{\partial u}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial u}{\partial p_i} \frac{\partial H}{\partial q_i} + \frac{\partial u}{\partial t} \\[6pt] &= [u, H] + \frac{\partial u}{\partial t}, \end{align}$$ where $$H$$ is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as $$\dot{q}_i = [q_i, H]$$ and $$\dot{p}_i = [p_i, H]$$.

Suppose $$u$$ is a constant of motion, then it must satisfy"$[H, u] = \frac{\partial u}{\partial t}.$"Moreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion.

In September 1925, Paul Dirac received proofs of a seminal paper by Werner Heisenberg on the new branch of physics known as quantum mechanics. Soon he realized that the key idea in Heisenberg's paper was the anti-commutativity of dynamical variables and remembered that the analogous mathematical construction in classical mechanics was Poisson brackets. He found the treatment he needed in E. T. Whittaker's Analytical Dynamics of Particles and Rigid Bodies.

Continuum mechanics and fluid flow
In 1821, using an analogy with elastic bodies, Claude-Louis Navier arrived at the basic equations of motion for viscous fluids, now identified as the Navier–Stokes equations. In 1829 Poisson independently obtained the same result. George Gabriel Stokes re-derived them in 1845 using continuum mechanics. Poisson, Augustin-Louis Cauchy, and Sophie Germain were the main contributors to the theory of elasticity in the nineteenth century. The calculus of variations was frequently used to solve problems.

Wave propagation
Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825).

Thermodynamics
In his work on heat conduction, Joseph Fourier maintained that the arbitrary function may be represented as an infinite trigonometric series and made explicit the possibility of expanding functions in terms of Bessel functions and Legendre polynomials, depending on the context of the problem. It took some time for his ideas to be accepted as his use of mathematics was less than rigorous. Although initially skeptical, Poisson adopted Fourier's method. From around 1815 he studied various problems in heat conduction. He published his Théorie mathématique de la chaleur in 1835.

During the early 1800s, Pierre-Simon de Laplace developed a sophisticated, if speculative, description of gases based on the old caloric theory of heat, to which younger scientists such as Poisson were less committed. A success for Laplace was his correction of Newton's formula for the speed of sound in air that gives satisfactory answers when compared with experiments. The Newton–Laplace formula makes use of the specific heats of gases at constant volume $$c_V$$and at constant pressure $$c_P$$. In 1823 Poisson redid his teacher's work and reached the same results without resorting to complex hypotheses previously employed by Laplace. In addition, by using the gas laws of Robert Boyle and Joseph Louis Gay-Lussac, Poisson obtained the equation for gases undergoing adiabatic changes, namely $$PV^{\gamma} = \text{constant}$$, where $$P$$ is the pressure of the gas, $$V$$ its volume, and $$\gamma = \frac{c_P}{c_V}$$.

Other works
Besides his many memoirs, Poisson published a number of treatises, most of which were intended to form part of a great work on mathematical physics, which he did not live to complete. Among these may be mentioned:
 * Nouvelle théorie de l'action capillaire (4to, 1831);
 * Recherches sur la probabilité des jugements en matières criminelles et matière civile (4to, 1837), all published at Paris.
 * A catalog of all of Poisson's papers and works can be found in Oeuvres complétes de François Arago, Vol. 2
 * Mémoire sur l'équilibre et le mouvement des corps élastiques (v. 8 in Mémoires de l'Académie Royale des Sciences de l'Institut de France, 1829), digitized copy from the Bibliothèque nationale de France
 * Recherches sur le Mouvement des Projectiles dans l'Air, en ayant égard a leur figure et leur rotation, et a l'influence du mouvement diurne de la terre (1839)

Interaction with Évariste Galois
After political activist Évariste Galois had returned to mathematics after his expulsion from the École Normale, Poisson asked him to submit his work on the theory of equations, which he did January 1831. In early July, Poisson declared Galois' work "incomprehensible," but encouraged Galois to "publish the whole of his work in order to form a definitive opinion." While Poisson's report was made before Galois' 14 July arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois vehemently decided against publishing his papers through the academy and instead publish them privately through his friend Auguste Chevalier. Yet Galois did not ignore Poisson's advice. He began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832, after which he was somehow persuaded to participate in what proved to be a fatal duel.

title: Siméon Denis Poisson


text:
The Mothman Prophecies

The Mothman Prophecies is a 1975 book by John Keel.

Synopsis
The book relates Keel's accounts of his investigation into alleged sightings of a large, winged creature known as Mothman in the vicinity of Point Pleasant, West Virginia, during 1966 and 1967. It combines these accounts with his theories about UFOs and various supernatural phenomena, ultimately connecting them to the collapse of the Silver Bridge across the Ohio River on December 15, 1967. Official investigations in 1971 determined the bridge collapse was caused by stress corrosion cracking in an eyebar in a suspension chain.

Reception
Kirkus Reviews wrote that the book featured Keel's theories that "ultraterrestrials" use some form of psychic power to create hallucinations such as Mothman and UFOs.

In the May/June 2002 issue of Skeptical Inquirer, journalist John C. Sherwood, a former business associate of UFO researcher Gray Barker, published an analysis of private letters between Keel and Barker during the period of Keel's investigation. In the article, "Gray Barker's Book of Bunk," Sherwood reported finding significant differences between what Keel wrote at the time of his investigation and what he wrote in his first book about the Mothman reports, raising questions about the book's accuracy. Sherwood also reported that Keel, who was well known for writing humorous and outrageous letters to friends and associates, would not assist him in clarifying the differences.

Film
The Mothman Prophecies was the inspiration for the 2002 film of the same name, starring Richard Gere.

title: The Mothman Prophecies