Evangelista Torricelli ( / ˌ t ɒr i ˈ tʃ ɛ l i / TORR -ee- CHEL -ee ; Italian: [evandʒeˈlista torriˈtʃɛlli] ; 15 October 1608 – 25 October 1647) was an Italian physicist and mathematician, and a student of Galileo . He is best known for his invention of the barometer , but is also known for his advances in optics and work on the method of indivisibles . The torr is named after him.
173-637: Torricelli was born on 15 October 1608 in Rome , the firstborn child of Gaspare Torricelli and Caterina Angetti. His family was from Faenza in the Province of Ravenna , then part of the Papal States . His father was a textile worker and the family was very poor. Seeing his talents, his parents sent him to be educated in Faenza, under the care of his uncle, Giacomo (James), a Camaldolese monk , who first ensured that his nephew
346-679: A 0 − L {\displaystyle a_{0}-L} , where L is the limit of the sequence, and in this way gives a proof of the formula for the sum of a geometric series. Torricelli developed further the method of indivisibles of Cavalieri . Many 17th century mathematicians learned of the method through Torricelli whose writing was more accessible than Cavalieri's. Several Italian Navy submarines were named after Evangelista Torricelli: His original manuscripts are preserved at Florence, Italy. The following have appeared in print: 1 Rome Rome ( Italian and Latin : Roma , pronounced [ˈroːma] )
519-424: A 0 , a 1 , a 2 , … {\displaystyle a_{0},a_{1},a_{2},\ldots } and showed the corresponding telescoping series ( a 0 − a 1 ) + ( a 1 − a 2 ) + ⋯ {\displaystyle (a_{0}-a_{1})+(a_{1}-a_{2})+\cdots } necessarily converges to
692-456: A civil war against the Senate and Pompey. After his victory, Caesar established himself as dictator for life . His assassination in 44 BC led to a second Triumvirate among Octavian (Caesar's grandnephew and heir), Mark Antony and Lepidus , and to a final civil war between Octavian and Antony. In 27 BC, Octavian was named Augustus and princeps , founding the principate ,
865-510: A diarchy between the princeps and the senate. Over time, the new monarch came to be known as the imperator (hence emperor ), meaning "commander". During the reign of Nero , two thirds of the city was ruined after the Great Fire of Rome , and the persecution of Christians commenced. Rome's empire reached its greatest expansion in the second century under the Emperor Trajan . Rome
1038-463: A senatore or patrizio . In the 12th century, this administration, like other European cities, evolved into the commune , a new form of social organisation controlled by the new wealthy classes. Pope Lucius II fought against the Roman commune, and the struggle was continued by his successor Pope Eugenius III : by this stage, the commune, allied with the aristocracy, was supported by Arnaldo da Brescia ,
1211-464: A series of civil wars between rival claimants to power resulted in the unification of the empire under Constantine the Great in 324. Hereditary succession was restored, but the east–west division was maintained. Constantine undertook a major reform of the bureaucracy, not by changing the structure but by rationalising the competencies of the several ministries. The so-called Edict of Milan of 313, actually
1384-434: A 50-meter bell tower, and much more so at the peak of a 1460-meter mountain. As we know now, the column's height fluctuates with atmospheric pressure at the same location, a fact which plays a key role in weather forecasting. Baseline changes in the column's height at different elevations, in turn, underlie the principle of the altimeter. Thus, this work laid the foundations for the modern concept of atmospheric pressure ,
1557-430: A broad band, or extremely low reflectivity at a single wavelength. Constructive interference in thin films can create a strong reflection of light in a range of wavelengths, which can be narrow or broad depending on the design of the coating. These films are used to make dielectric mirrors , interference filters , heat reflectors , and filters for colour separation in colour television cameras. This interference effect
1730-619: A changing index of refraction; this principle allows for lenses and the focusing of light. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n 1 and another medium with index of refraction n 2 . In such situations, Snell's Law describes the resulting deflection of the light ray: n 1 sin θ 1 = n 2 sin θ 2 {\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}} where θ 1 and θ 2 are
1903-493: A common paraboloid. This envelope became known as the parabola di sicurezza ( parabola of safety )." Torricelli gave the first scientific description of the cause of wind : ... winds are produced by differences of air temperature, and hence density, between two regions of the earth. Torricelli is also famous for the discovery of the Torricelli's trumpet (also - perhaps more often - known as Gabriel's Horn ) whose surface area
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#17328553323402076-399: A converging lens has positive focal length, while a diverging lens has negative focal length. Smaller focal length indicates that the lens has a stronger converging or diverging effect. The focal length of a simple lens in air is given by the lensmaker's equation . Ray tracing can be used to show how images are formed by a lens. For a thin lens in air, the location of the image is given by
2249-562: A crusade against the Colonna family and, in 1300, called for the first Jubilee of Christianity , which brought millions of pilgrims to Rome. However, his hopes were crushed by the French king Philip the Fair , who took him prisoner and held him hostage for three days at Anagni . The Pope was able to return to Rome, but died a month later, it was said of shock and grief. Afterwards, a new pope faithful to
2422-419: A fragment of a letter from his co-emperor Licinius to the governors of the eastern provinces, granted freedom of worship to everyone, including Christians, and ordered the restoration of confiscated church properties upon petition to the newly created vicars of dioceses. He funded the building of several churches and allowed clergy to act as arbitrators in civil suits (a measure that did not outlast him but which
2595-420: A fringe hypothesis. Traditional stories handed down by the ancient Romans themselves explain the earliest history of their city in terms of legend and myth . The most familiar of these myths, and perhaps the most famous of all Roman myths , is the story of Romulus and Remus , the twins who were suckled by a she-wolf . They decided to build a city, but after an argument, Romulus killed his brother and
2768-431: A height of 10 metres. After Galileo's death, Torricelli proposed, rather, that we live in a "sea of air" that exerts a pressure analogous in many ways to the pressure of water on submerged objects. According to this hypothesis, at sea level, the air in the atmosphere has weight that roughly equals the weight of a 10-meter column of water. When a suction pump creates a vacuum inside a tube, the atmosphere no longer pushes on
2941-511: A large area in Rome, and the city has one of the largest areas of green space among European capitals. The most notable part of this green space is represented by the large number of villas and landscaped gardens created by the Italian aristocracy. While most of the parks surrounding the villas were destroyed during the building boom of the late 19th century, some of them remain. The most notable of these are
3114-619: A monk who was a religious and social reformer. After the pope's death, Arnaldo was taken prisoner by Adrianus IV , which marked the end of the commune's autonomy. Under Pope Innocent III , whose reign marked the apogee of the papacy, the commune liquidated the senate, and replaced it with a Senatore , who was subject to the pope. In this period, the papacy played a role of secular importance in Western Europe , often acting as arbitrators between Christian monarchs and exercising additional political powers. In 1266, Charles of Anjou , who
3287-503: A number of telescopes and simple microscopes; several large lenses, engraved with his name, are still preserved in Florence . On 11 June 1644, he famously wrote in a letter to Michelangelo Ricci : Noi viviamo sommersi nel fondo d'un pelago d'aria. (We live submerged at the bottom of an ocean of air.) However his work on the cycloid involved him in a controversy with Gilles de Roberval , who accused him of plagiarizing his earlier solution of
3460-611: A population of 4,355,725 residents, is the most populous metropolitan city in Italy. Its metropolitan area is the third-most populous within Italy. Rome is located in the central-western portion of the Italian Peninsula , within Lazio ( Latium ), along the shores of the Tiber Valley . Vatican City (the smallest country in the world and headquarters of the worldwide Catholic Church under
3633-496: A private arrangement. Because of this, Torricelli was exposed to experiments funded by Pope Urban VIII . While living in Rome, Torricelli became also the student of the mathematician Bonaventura Cavalieri , with whom he became great friends. It was in Rome that Torricelli also became friends with two other students of Castelli, Raffaello Magiotti and Antonio Nardi . Galileo referred to Torricelli, Magiotti, and Nardi affectionately as his "triumvirate" in Rome. In 1632, shortly after
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#17328553323403806-477: A single scalar quantity to represent the electric field of the light wave, rather than using a vector model with orthogonal electric and magnetic vectors. The Huygens–Fresnel equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of superposition of waves. The Kirchhoff diffraction equation , which
3979-522: A single point on the image, while chromatic aberration occurs because the index of refraction of the lens varies with the wavelength of the light. In physical optics, light is considered to propagate as waves. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The speed of light waves in air is approximately 3.0×10 m/s (exactly 299,792,458 m/s in vacuum ). The wavelength of visible light waves varies between 400 and 700 nm, but
4152-452: A small leak at the bottom and y is the depth of the water at time t , then for some constant k > 0. The concept of center of gravity was discovered by Archimedes. Torricelli, following in his footsteps, discovered an important new principle, Torricelli’s principle, which says: if any number of bodies be so connected that, by their motion, their centre of gravity can neither ascend nor descend, then those bodies are in equilibrium. This
4325-437: A spectrum. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton. Some media have an index of refraction which varies gradually with position and, therefore, light rays in the medium are curved. This effect is responsible for mirages seen on hot days: a change in index of refraction air with height causes light rays to bend, creating the appearance of specular reflections in
4498-535: A statue of Torricelli was created in 1868 in gratitude for all that Torricelli had done in advancing science during his short lifetime. The asteroid 7437 Torricelli and a crater on the Moon were named in his honour. The mountain range Torricelli Mountains on New Guinea carry his name. In 1830, botanist Augustin Pyramus de Candolle published Torricellia , which is a genus of flowering plants from Asia belonging to
4671-463: A theoretical and practical problem: a suction pump could only raise water up to a height of 10 metres (34 ft) (as recounted in Galileo's Two New Sciences ). In the early 1600s, Torricelli's teacher, Galileo, argued that suction pumps were able to draw water from a well because of the "force of vacuum." This argument, however, failed to explain the fact that suction pumps could only raise water to
4844-465: A thickness of one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550 nm. More complex designs using multiple layers can achieve low reflectivity over
5017-426: A unit of pressure used in vacuum measurements, is named after him. Torricelli also discovered a law, regarding the speed of a fluid flowing out of an opening, which was later shown to be a particular case of Bernoulli's principle . He found that water leaks out a small hole in the bottom of a container at a rate proportional to the square root of the depth of the water. So if the container is an upright cylinder with
5190-476: A variety of technologies and everyday objects, including mirrors , lenses , telescopes , microscopes , lasers , and fibre optics . Optics began with the development of lenses by the ancient Egyptians and Mesopotamians . The earliest known lenses, made from polished crystal , often quartz , date from as early as 2000 BC from Crete (Archaeological Museum of Heraclion, Greece). Lenses from Rhodes date around 700 BC, as do Assyrian lenses such as
5363-607: A very rare occurrence (the most recent ones were in 2018, 2012 and 1986). Optics Optics is the branch of physics that studies the behaviour and properties of light , including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible , ultraviolet , and infrared light. Light is a type of electromagnetic radiation , and other forms of electromagnetic radiation such as X-rays , microwaves , and radio waves exhibit similar properties. Most optical phenomena can be accounted for by using
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5536-539: A warrior pope, to Alexander VI , immoral and nepotist , from Julius II , soldier and patron, to Leo X , who gave his name to this period ("the century of Leo X"), all devoted their energy to the greatness and the beauty of the Eternal City and to the patronage of the arts. During those years, the centre of the Italian Renaissance moved to Rome from Florence. Majestic works, as the new Saint Peter's Basilica ,
5709-525: A wide range of scientific topics, and discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light, basing it on the works of Aristotle and Platonism. Grosseteste's most famous disciple, Roger Bacon , wrote works citing a wide range of recently translated optical and philosophical works, including those of Alhazen, Aristotle, Avicenna , Averroes , Euclid, al-Kindi, Ptolemy, Tideus, and Constantine
5882-407: Is infinite , but whose volume is finite. This was seen as an "incredible" paradox by many at the time, including Torricelli himself, and prompted a fierce controversy about the nature of infinity, also involving the philosopher Hobbes . Torricelli was also a pioneer in the area of infinite series. In his De dimensione parabolae of 1644, Torricelli considered a decreasing sequence of positive terms
6055-591: Is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation. Some phenomena depend on light having both wave-like and particle-like properties . Explanation of these effects requires quantum mechanics . When considering light's particle-like properties,
6228-427: Is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. In the absence of nonlinear effects,
6401-458: Is a wide consensus that the city developed gradually through the aggregation (" synoecism ") of several villages around the largest one, placed above the Palatine. This aggregation was facilitated by the increase of agricultural productivity above the subsistence level , which also allowed the establishment of secondary and tertiary activities . These, in turn, boosted the development of trade with
6574-405: Is above 21 °C (70 °F) during the day and 9 °C (48 °F) at night. In the coldest month, January, the average temperature is 12.6 °C (54.7 °F) during the day and 2.1 °C (35.8 °F) at night. In the warmest month, August, the average temperature is 31.7 °C (89.1 °F) during the day and 17.3 °C (63.1 °F) at night. December, January and February are
6747-566: Is also a notable pine wood at Castelfusano , near Ostia. Rome also has a number of regional parks of much more recent origin, including the Pineto Regional Park and the Appian Way Regional Park. There are also nature reserves at Marcigliana and at Tenuta di Castelporziano. Rome has a Mediterranean climate ( Köppen climate classification : Csa ), with hot, dry summers and mild, humid winters. Its average annual temperature
6920-605: Is another large green space: it has few trees but is overlooked by the Palatine and the Rose Garden ('roseto comunale'). Nearby is the lush Villa Celimontana , close to the gardens surrounding the Baths of Caracalla. The Villa Borghese garden is the best known large green space in Rome, with famous art galleries among its shaded walks. Overlooking Piazza del Popolo and the Spanish Steps are the gardens of Pincio and Villa Medici . There
7093-548: Is based on the amount of pork, 3,629,000 lbs. distributed to poorer Romans during five winter months at the rate of five Roman lbs per person per month, enough for 145,000 persons or 1/4 or 1/3 of the total population. Grain distribution to 80,000 ticket holders at the same time suggests 400,000 (Augustus set the number at 200,000 or one-fifth of the population). After the fall of the Western Roman Empire in AD ;476, Rome
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7266-486: Is considered to travel in straight lines, while in physical optics, light is considered as an electromagnetic wave. Geometrical optics can be viewed as an approximation of physical optics that applies when the wavelength of the light used is much smaller than the size of the optical elements in the system being modelled. Geometrical optics , or ray optics , describes the propagation of light in terms of "rays" which travel in straight lines, and whose paths are governed by
7439-484: Is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens–Fresnel principle can be found in the articles on diffraction and Fraunhofer diffraction . More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect
7612-446: Is essentially a version of the principle of virtual work. This principle was later used by Christiaan Huygens to study pendulum motion. Torricelli studied projectiles and how they traveled through the air. "Perhaps his most notable achievement in the field of projectiles was to establish for the first time the idea of an envelope : projectiles sent out at [...] the same speed in all directions trace out parabolas which are all tangent to
7785-555: Is generally thought the population of the city until AD 300 was 1 million (estimates range from 2 million to 750,000) declining to 750–800,000 in AD 400, then 450–500,000 in AD 450 and down to 80–100,000 in AD 500 (though it may have been twice this). The Bishop of Rome, called the Pope , was important since the early days of Christianity because of the martyrdom of both the apostles Peter and Paul there. The Bishops of Rome were also seen (and still are seen by Catholics) as
7958-412: Is possible however that Torricelli carried out the mercury experiment first, and then formulated his sea of air hypothesis). In 1643, Torricelli filled a meter-long tube (with one end sealed off) with mercury —thirteen times denser than water—and set the open end of the tube into a basin of the liquid metal and raised the sealed end so the tube stood vertically. The mercury level in the tube fell until it
8131-441: Is possible that the name Romulus was actually derived from Rome itself. As early as the 4th century, there have been alternative theories proposed on the origin of the name Roma . Several hypotheses have been advanced focusing on its linguistic roots which however remain uncertain: Rome has also been called in ancient times simply "Urbs" (central city), from urbs roma , or identified with its ancient Roman initialism of SPQR ,
8304-624: Is the capital city of Italy . It is also the capital of the Lazio region , the centre of the Metropolitan City of Rome Capital , and a special comune (municipality) named Comune di Roma Capitale . With 2,860,009 residents in 1,285 km (496.1 sq mi), Rome is the country's most populated comune and the third most populous city in the European Union by population within city limits. The Metropolitan City of Rome, with
8477-409: Is to the lens, the further the image is from the lens. With diverging lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at a spot one focal length in front of the lens. This is the lens's front focal point. Rays from an object at a finite distance are associated with a virtual image that is closer to the lens than the focal point, and on
8650-476: The Book of Optics ( Kitab al-manazir ) in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment. He rejected the "emission theory" of Ptolemaic optics with its rays being emitted by the eye, and instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and then entered
8823-632: The Age of Enlightenment , new ideas reached the Eternal City, where the papacy supported archaeological studies and improved the people's welfare. But not everything went well for the Church during the Counter-Reformation. There were setbacks in the attempts to assert the Church's power, a notable example being in 1773 when Pope Clement XIV was forced by secular powers to have the Jesuit order suppressed . The rule of
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#17328553323408996-520: The Dominate , derived from his title of dominus ("lord"). His most marked feature was the unprecedented intervention of the State down to the city level: whereas the State had submitted a tax demand to a city and allowed it to allocate the charges, from his reign the State did this down to the village level. In a vain attempt to control inflation, he imposed price controls which did not last. Diocletian divided
9169-1163: The Food and Agriculture Organization (FAO), the World Food Programme (WFP), the International Fund for Agricultural Development (IFAD) and the UN System Network on Rural Development and Food Security . The city also hosts the European Union (EU) Delegation to the United Nations (UN) and the Secretariat of the Parliamentary Assembly of the Union for the Mediterranean (UfM) as well as the headquarters of several Italian multinational companies, such as Eni , Enel , TIM , Leonardo , and banks such as BNL . Numerous companies are based within Rome's EUR business district, such as
9342-654: The Gauls , Osci - Samnites and the Greek colony of Taranto , allied with Pyrrhus , king of Epirus ) whose result was the conquest of the Italian peninsula , from the central area up to Magna Graecia . The 3rd and 4th century BC saw the establishment of Roman hegemony over the Mediterranean and the Balkans through the three Punic Wars (264–146 BC) fought against Carthage and
9515-607: The Nimrud lens . The ancient Romans and Greeks filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world . The word optics comes from the ancient Greek word ὀπτική , optikē ' appearance, look ' . Greek philosophy on optics broke down into two opposing theories on how vision worked,
9688-503: The Roman Empire , and is regarded by many as the first-ever Imperial city and metropolis . It was first called The Eternal City ( Latin : Urbs Aeterna ; Italian : La Città Eterna ) by the Roman poet Tibullus in the 1st century BC, and the expression was also taken up by Ovid , Virgil , and Livy . Rome is also called Caput Mundi (Capital of the World). After the fall of
9861-586: The San Lorenzo district was subject to Allied bombing raids , resulting in about 3,000 fatalities and 11,000 injuries, of whom another 1,500 died. Mussolini was arrested on 25 July 1943 . On the date of the Italian Armistice 8 September 1943 the city was occupied by the Germans. Allied bombing raids continued throughout 1943 and extended into 1944. Rome was liberated on 4 June 1944. Rome developed greatly after
10034-480: The Sapienza University of Rome ). Castelli was a student of Galileo Galilei . "Benedetto Castelli made experiments on running water (1628), and he was entrusted by Pope Urban VIII with hydraulic undertakings." There is no actual evidence that Torricelli was enrolled at the university. It is almost certain that Torricelli was taught by Castelli. In exchange he worked for him as his secretary from 1626 to 1632 in
10207-638: The Sistine Chapel and Ponte Sisto (the first bridge to be built across the Tiber since antiquity, although on Roman foundations) were created. To accomplish that, the Popes engaged the best artists of the time, including Michelangelo , Perugino , Raphael , Ghirlandaio , Luca Signorelli , Botticelli , and Cosimo Rosselli . The period was also infamous for papal corruption, with many Popes fathering children, and engaging in nepotism and simony . The corruption of
10380-614: The University of Pisa . Right before the appointment, Torricelli was considering returning to Rome because of there being nothing left for him in Florence, where he had invented the barometer . In this new role he solved some of the great mathematical problems of the day, such as finding a cycloid 's area and center of gravity. As a result of this study, he wrote the book the Opera Geometrica in which he described his observations. The book
10553-571: The Villa Borghese , Villa Ada , and Villa Doria Pamphili . Villa Doria Pamphili is west of the Gianicolo hill, comprising some 1.8 km (0.7 sq mi ). The Villa Sciarra is on the hill, with playgrounds for children and shaded walking areas. In the nearby area of Trastevere, the Orto Botanico (Botanical Garden) is a cool and shady green space. The old Roman hippodrome (Circus Maximus)
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#173285533234010726-398: The classical electromagnetic description of light, however complete electromagnetic descriptions of light are often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics , treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics
10899-449: The emission theory , the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus . Some hundred years later, Euclid (4th–3rd century BC) wrote a treatise entitled Optics where he linked vision to geometry , creating geometrical optics . He based his work on Plato's emission theory wherein he described
11072-568: The founding of Rome at around 753 BC, the site has been inhabited for much longer, making it a major human settlement for over three millennia and one of the oldest continuously occupied cities in Europe. The city's early population originated from a mix of Latins , Etruscans , and Sabines . Eventually, the city successively became the capital of the Roman Kingdom , the Roman Republic and
11245-468: The intromission theory and the emission theory . The intromission approach saw vision as coming from objects casting off copies of themselves (called eidola) that were captured by the eye. With many propagators including Democritus , Epicurus , Aristotle and their followers, this theory seems to have some contact with modern theories of what vision really is, but it remained only speculation lacking any experimental foundation. Plato first articulated
11418-448: The superposition principle , which is a wave-like property not predicted by Newton's corpuscle theory. This work led to a theory of diffraction for light and opened an entire area of study in physical optics. Wave optics was successfully unified with electromagnetic theory by James Clerk Maxwell in the 1860s. The next development in optical theory came in 1899 when Max Planck correctly modelled blackbody radiation by assuming that
11591-463: The surface normal , a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays and the normal lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection . For flat mirrors , the law of reflection implies that images of objects are upright and
11764-594: The western and eastern empires respectively. The seat of government in the Western Roman Empire was transferred to Ravenna in 408, but from 450 the emperors mostly resided in Rome. Rome, which had lost its central role in the administration of the empire, was sacked in 410 by the Visigoths led by Alaric I , but very little physical damage was done, most of which was repaired. What could not be so easily replaced were portable items such as artwork in precious metals and items for domestic use (loot). The popes embellished
11937-506: The African . Bacon was able to use parts of glass spheres as magnifying glasses to demonstrate that light reflects from objects rather than being released from them. The first wearable eyeglasses were invented in Italy around 1286. This was the start of the optical industry of grinding and polishing lenses for these "spectacles", first in Venice and Florence in the thirteenth century, and later in
12110-525: The Balkans made serious uncoordinated incursions that were more like giant raiding parties rather than attempts to settle. The Persian Empire invaded from the east several times during the 230s to 260s but were eventually defeated. The civil wars ended in 285 with the final victory of Diocletian , who undertook the restoration of the State. He ended the Principate and introduced a new authoritarian model known as
12283-568: The Church. Under the popes from Pius IV to Sixtus V , Rome became the centre of a reformed Catholicism and saw the building of new monuments which celebrated the papacy. The popes and cardinals of the 17th and early 18th centuries continued the movement by having the city's landscape enriched with baroque buildings. This was another nepotistic age; the new aristocratic families ( Barberini , Pamphili , Chigi , Rospigliosi , Altieri , Odescalchi ) were protected by their respective popes, who built huge baroque buildings for their relatives. During
12456-608: The Empire in the west , which marked the beginning of the Middle Ages , Rome slowly fell under the political control of the Papacy , and in the 8th century, it became the capital of the Papal States , which lasted until 1870. Beginning with the Renaissance , almost all popes since Nicholas V (1447–1455) pursued a coherent architectural and urban programme over four hundred years, aimed at making
12629-467: The French was elected, and the papacy was briefly relocated to Avignon (1309–1377). During this period Rome was neglected, until a plebeian man, Cola di Rienzo , came to power. An idealist and a lover of ancient Rome, Cola dreamed about a rebirth of the Roman Empire: after assuming power with the title of Tribuno , his reforms were rejected by the populace. Forced to flee, Cola returned as part of
12802-559: The Gothic siege of 537, the population dropped to 30,000 but had risen to 90,000 by the papacy of Gregory the Great . The population decline coincided with the general collapse of urban life in the West in the fifth and sixth centuries, with few exceptions. Subsidized state grain distributions to the poorer members of society continued right through the sixth century and probably prevented the population from falling further. The figure of 450,000–500,000
12975-676: The Great (which by then was in a dilapidated state) was demolished and a new one begun. The city hosted artists like Ghirlandaio , Perugino , Botticelli and Bramante , who built the temple of San Pietro in Montorio and planned a great project to renovate the Vatican . Raphael, who in Rome became one of the most famous painters of Italy, created frescoes in the Villa Farnesina , the Raphael's Rooms , plus many other famous paintings. Michelangelo started
13148-399: The Greek colonies of southern Italy (mainly Ischia and Cumae ). These developments, which according to archaeological evidence took place during the mid-eighth century BC, can be considered as the "birth" of the city. Despite recent excavations at the Palatine hill, the view that Rome was founded deliberately in the middle of the eighth century BC, as the legend of Romulus suggests, remains
13321-548: The Huygens–Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns. Interferometry is the science of measuring these patterns, usually as a means of making precise determinations of distances or angular resolutions . The Michelson interferometer
13494-531: The Italian Republic. In 2019, Rome was the 14th most visited city in the world, with 8.6 million tourists, the third most visited city in the European Union, and the most popular tourist destination in Italy. Its historic centre is listed by UNESCO as a World Heritage Site . The host city for the 1960 Summer Olympics , Rome is also the seat of several specialised agencies of the United Nations , such as
13667-1014: The Kings was built on seven hills: the Aventine Hill , the Caelian Hill , the Capitoline Hill , the Esquiline Hill , the Palatine Hill , the Quirinal Hill , and the Viminal Hill . Modern Rome is also crossed by another river, the Aniene , which flows into the Tiber north of the historic centre. Although the city centre is about 24 km (15 mi) inland from the Tyrrhenian Sea , the city territory extends to
13840-518: The Papal States were reconstituted by a decision of the Congress of Vienna of 1814. In 1849, a second Roman Republic was proclaimed during a year of revolutions in 1848 . Two of the most influential figures of the Italian unification , Giuseppe Mazzini and Giuseppe Garibaldi , fought for the short-lived republic. Rome then became the focus of hopes of Italian reunification after the rest of Italy
14013-549: The Popes and the huge expenses for their building projects led, in part, to the Reformation and, in turn, the Counter-Reformation . Under extravagant and rich popes, Rome was transformed into a centre of art, poetry, music, literature, education and culture. Rome became able to compete with other major European cities of the time in terms of wealth, grandeur, the arts, learning and architecture. The Renaissance period changed
14186-560: The Popes was interrupted by the short-lived Roman Republic (1798–1800), which was established under the influence of the French Revolution . The Papal States were restored in June 1800, but during Napoleon 's reign Rome was annexed as a Département of the French Empire : first as Département du Tibre (1808–1810) and then as Département Rome (1810–1814). After the fall of Napoleon,
14359-496: The Romans expelled the last king from their city and established an oligarchic republic led by two annually-elected consuls . Rome then began a period characterised by internal struggles between patricians (aristocrats) and plebeians (small landowners), and by constant warfare against the populations of central Italy: Etruscans, Latins, Volsci , Aequi , and Marsi . After becoming master of Latium , Rome led several wars (against
14532-569: The Third Century , during which numerous generals fought for power and the central authority in Rome weakened dramatically. Around the same time, the Plague of Cyprian ( c. 250–270) afflicted the Mediterranean. Instability caused economic deterioration, and there was a rapid rise in inflation as the government debased the currency in order to meet expenses. The Germanic tribes along the Rhine and north of
14705-484: The amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in destructive interference and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect. Since
14878-489: The angle of incidence. Plutarch (1st–2nd century AD) described multiple reflections on spherical mirrors and discussed the creation of magnified and reduced images, both real and imaginary, including the case of chirality of the images. During the Middle Ages , Greek ideas about optics were resurrected and extended by writers in the Muslim world . One of the earliest of these was Al-Kindi ( c. 801 –873) who wrote on
15051-435: The angles between the normal (to the interface) and the incident and refracted waves, respectively. The index of refraction of a medium is related to the speed, v , of light in that medium by n = c / v , {\displaystyle n=c/v,} where c is the speed of light in vacuum . Snell's Law can be used to predict the deflection of light rays as they pass through linear media as long as
15224-403: The birth of the conclave . In this period the city was also shattered by continuous fights between the aristocratic families: Annibaldi , Caetani , Colonna , Orsini , Conti , nested in their fortresses built above ancient Roman edifices, fought each other to control the papacy. Pope Boniface VIII , born Caetani, was the last pope to fight for the church's universal domain ; he proclaimed
15397-510: The city centre in order to build wide avenues and squares which were supposed to celebrate the fascist regime and the resurgence and glorification of classical Rome. The interwar period saw a rapid growth in the city's population which surpassed one million inhabitants soon after 1930. During World War II, due to the art treasuries and the presence of the Vatican, Rome largely escaped the tragic destiny of other European cities. However, on 19 July 1943,
15570-530: The city the artistic and cultural centre of the world. In this way, Rome first became one of the major centres of the Renaissance and then became the birthplace of both the Baroque style and Neoclassicism . Famous artists, painters, sculptors, and architects made Rome the centre of their activity, creating masterpieces throughout the city. In 1871, Rome became the capital of the Kingdom of Italy , which, in 1946, became
15743-557: The city took his name. According to the Roman annalists , this happened on 21 April 753 BC. This legend had to be reconciled with a dual tradition, set earlier in time, that had the Trojan refugee Aeneas escape to Italy and found the line of Romans through his son Iulus , the namesake of the Julio-Claudian dynasty . This was accomplished by the Roman poet Virgil in the first century BC. In addition, Strabo mentions an older story, that
15916-605: The city was affected by the divisions which rocked the Church. In 1418, the Council of Constance settled the Western Schism , and a Roman pope, Martin V , was elected. This brought to Rome a century of internal peace, which marked the beginning of the Renaissance . The ruling popes until the first half of the 16th century, from Nicholas V , founder of the Vatican Library , to Pius II , humanist and literate, from Sixtus IV ,
16089-554: The city was an Arcadian colony founded by Evander . Strabo also writes that Lucius Coelius Antipater believed that Rome was founded by Greeks. After the foundation by Romulus according to a legend, Rome was ruled for a period of 244 years by a monarchical system , initially with sovereigns of Latin and Sabine origin, later by Etruscan kings. The tradition handed down seven kings: Romulus , Numa Pompilius , Tullus Hostilius , Ancus Marcius , Tarquinius Priscus , Servius Tullius and Lucius Tarquinius Superbus . In 509 BC,
16262-466: The city with large basilicas, such as Santa Maria Maggiore (with the collaboration of the emperors). The population of the city had fallen from 800,000 to 450–500,000 by the time the city was sacked in 455 by Genseric , king of the Vandals . The weak emperors of the fifth century could not stop the decay, leading to the deposition of Romulus Augustus , who resided on Ravenna, on 4 September 476. This marked
16435-455: The coldest months, with a daily mean temperature of approximately 8 °C (46 °F). Temperatures during these months generally vary between 10 and 15 °C (50 and 59 °F) during the day and between 3 and 5 °C (37 and 41 °F) at night, with colder or warmer spells occurring frequently. Snowfall is rare but not unheard of, with light snow or flurries occurring on some winters, generally without accumulation, and major snowfalls on
16608-569: The decoration of the ceiling of the Sistine Chapel and executed the famous statue of the Moses for the tomb of Julius II. Its economy was rich, with the presence of several Tuscan bankers, including Agostino Chigi , who was a friend of Raphael and a patron of arts. Before his early death, Raphael also promoted for the first time the preservation of the ancient ruins. The War of the League of Cognac caused
16781-449: The distance (as if on the surface of a pool of water). Optical materials with varying indexes of refraction are called gradient-index (GRIN) materials. Such materials are used to make gradient-index optics . For light rays travelling from a material with a high index of refraction to a material with a low index of refraction, Snell's law predicts that there is no θ 2 when θ 1 is large. In this case, no transmission occurs; all
16954-541: The empire in 286, ruling over the eastern half from Nicomedia , while his co-emperor Maximian ruled the western half from Mediolanum (when not on the move). The empire was further divided in 293, when Diocletian named two caesar , one for each augustus (emperor). Diocletian tried to turn into a system of non-dynastic succession, similar to the Antonine dynasty. Upon abdication in 305, both caesars succeeded and they, in turn, appointed two colleagues for themselves. However,
17127-468: The end of the Western Roman Empire and, for many historians, the beginning of the Middle Ages . The decline of the city's population was caused by the loss of grain shipments from North Africa, from 440 onward, and the unwillingness of the senatorial class to maintain donations to support a population that was too large for the resources available. Even so, strenuous efforts were made to maintain
17300-400: The entourage of Cardinal Albornoz , who was charged with restoring the Church's power in Italy. Back in power for a short time, Cola was soon lynched by the populace, and Albornoz took possession of the city. In 1377, Rome became the seat of the papacy again under Gregory XI . The return of the pope to Rome in that year unleashed the Western Schism (1377–1418), and for the next forty years,
17473-426: The exchange of energy between light and matter only occurred in discrete amounts he called quanta . In 1905, Albert Einstein published the theory of the photoelectric effect that firmly established the quantization of light itself. In 1913, Niels Bohr showed that atoms could only emit discrete amounts of energy, thus explaining the discrete lines seen in emission and absorption spectra . The understanding of
17646-525: The eye, although he was unable to correctly explain how the eye captured the rays. Alhazen's work was largely ignored in the Arabic world but it was anonymously translated into Latin around 1200 A.D. and further summarised and expanded on by the Polish monk Witelo making it a standard text on optics in Europe for the next 400 years. In the 13th century in medieval Europe, English bishop Robert Grosseteste wrote on
17819-584: The face of Rome dramatically, with works like the Pietà by Michelangelo and the frescoes of the Borgia Apartments . Rome reached the highest point of splendour under Pope Julius II (1503–1513) and his successors Leo X and Clement VII , both members of the Medici family . In this twenty-year period, Rome became one of the greatest centres of art in the world. The old St. Peter's Basilica built by Emperor Constantine
17992-423: The failed attempt of social reform of the populares Tiberius and Gaius Gracchus , and the war against Jugurtha , there was a civil war from which the general Sulla emerged victorious. A major slave revolt under Spartacus followed, and then the establishment of the first Triumvirate with Caesar , Pompey and Crassus . The conquest of Gaul made Caesar immensely powerful and popular, which led to
18165-586: The family Torricelliaceae . They were named in Evangelista Torricelli's honour. The perusal of Galileo's Two New Sciences (1638) inspired Torricelli with many developments of the mechanical principles there set forth, which he embodied in a treatise De motu (printed amongst his Opera geometrica , 1644). Its communication by Castelli to Galileo in 1641, with a proposal that Torricelli should reside with him, led to Torricelli traveling to Florence , where he met Galileo, and acted as his amanuensis during
18338-535: The feud between the two lasted until Hooke's death. In 1704, Newton published Opticks and, at the time, partly because of his success in other areas of physics, he was generally considered to be the victor in the debate over the nature of light. Newtonian optics was generally accepted until the early 19th century when Thomas Young and Augustin-Jean Fresnel conducted experiments on the interference of light that firmly established light's wave nature. Young's famous double slit experiment showed that light followed
18511-435: The first barometer , an instrument that would later play a key role in weather forecasting, and the first pressure altimeter , which measures altitude and is often used in hiking, climbing, skiing, and aviation. The solution to the suction pump puzzle and the discovery of the principle of the barometer and altimeter have perpetuated Torricelli's fame with terms such as "Torricellian tube" and "Torricellian vacuum". The torr ,
18684-603: The first half of the first century (under Tiberius ) and popularised by his apostles through the empire and beyond. The Antonine age is considered the zenith of the Empire, whose territory ranged from the Atlantic Ocean to the Euphrates and from Britain to Egypt . After the end of the Severan dynasty in AD 235, the Empire entered into a 50-year period known as the Crisis of
18857-674: The first plunder of the city in more than five hundred years since the previous sack ; in 1527, the Landsknechts of Emperor Charles V sacked the city , bringing an abrupt end to the golden age of the Renaissance in Rome. Beginning with the Council of Trent in 1545, the Church began the Counter-Reformation in response to the Reformation, a large-scale questioning of the Church's authority on spiritual matters and governmental affairs. This loss of confidence led to major shifts of power away from
19030-474: The focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration . Curved mirrors can form images with a magnification greater than or less than one, and the magnification can be negative, indicating that the image is inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction occurs when light travels through an area of space that has
19203-411: The gloss of surfaces such as mirrors, which reflect light in a simple, predictable way. This allows for the production of reflected images that can be associated with an actual ( real ) or extrapolated ( virtual ) location in space. Diffuse reflection describes non-glossy materials, such as paper or rock. The reflections from these surfaces can only be described statistically, with the exact distribution of
19376-642: The governance of the Holy See ) is an independent country inside the city boundaries of Rome, the only existing example of a country within a city. Rome is often referred to as the City of Seven Hills due to its geographic location, and also as the "Eternal City". Rome is generally considered to be the cradle of Western civilization and Western Christian culture , and the centre of the Catholic Church . Rome's history spans 28 centuries. While Roman mythology dates
19549-471: The help of the plebs (urban lower class) to gain power. In the same period, the bankruptcy of the small farmers and the establishment of large slave estates caused large-scale migration to the city. The continuous warfare led to the establishment of a professional army, which turned out to be more loyal to its generals than to the republic. Because of this, in the late 2nd and early 1st century BC there were several conflicts both abroad and internally: after
19722-416: The incident rays came. This is called retroreflection . Mirrors with curved surfaces can be modelled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces , parallel rays incident on the mirror produce reflected rays that converge at a common focus . Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing
19895-418: The indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. In most materials, the index of refraction varies with the frequency of the light, known as dispersion . Taking this into account, Snell's Law can be used to predict how a prism will disperse light into
20068-436: The interaction between light and matter that followed from these developments not only formed the basis of quantum optics but also was crucial for the development of quantum mechanics as a whole. The ultimate culmination, the theory of quantum electrodynamics , explains all optics and electromagnetic processes in general as the result of the exchange of real and virtual photons. Quantum optics gained practical importance with
20241-426: The interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light. Numerical modeling techniques such as the finite element method , the boundary element method and the transmission-line matrix method can be used to model
20414-477: The invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle making centres in the Netherlands. In the early 17th century, Johannes Kepler expanded on geometric optics in his writings, covering lenses, reflection by flat and curved mirrors, the principles of pinhole cameras , inverse-square law governing the intensity of light, and
20587-491: The inventions of the maser in 1953 and of the laser in 1960. Following the work of Paul Dirac in quantum field theory , George Sudarshan , Roy J. Glauber , and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light. Classical optics is divided into two main branches: geometrical (or ray) optics and physical (or wave) optics. In geometrical optics, light
20760-504: The laws of reflection and refraction at interfaces between different media. These laws were discovered empirically as far back as 984 AD and have been used in the design of optical components and instruments from then until the present day. They can be summarised as follows: When a ray of light hits the boundary between two transparent materials, it is divided into a reflected and a refracted ray. The laws of reflection and refraction can be derived from Fermat's principle which states that
20933-449: The light is modelled as a collection of particles called " photons ". Quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy , various engineering fields, photography , and medicine (particularly ophthalmology and optometry , in which it is called physiological optics). Practical applications of optics are found in
21106-422: The light is reflected. This phenomenon is called total internal reflection and allows for fibre optics technology. As light travels down an optical fibre, it undergoes total internal reflection allowing for essentially no light to be lost over the length of the cable. A device that produces converging or diverging light rays due to refraction is known as a lens . Lenses are characterized by their focal length :
21279-590: The luxury fashion house Fendi located in the Palazzo della Civiltà Italiana . The presence of renowned international brands in the city has made Rome an important centre of fashion and design, and the Cinecittà Studios have been the set of many Academy Award –winning movies. According to the Ancient Romans' founding myth , the name Roma came from the city's founder and first king , Romulus . However, it
21452-483: The major part of the propositions and sundry problems which were gathered together by Viviani after Torricelli's death. This early work owes much to the study of the classics." Sixty-eight years after Torricelli had died, his genius still filled his contemporaries with admiration, as evidenced by the anagram below the frontispice of Lezioni accademiche d'Evangelista Torricelli published in 1715: En virescit Galileus alter, meaning "Here blossoms another Galileo." In Faenza,
21625-443: The mathematical rules of perspective and described the effects of refraction qualitatively, although he questioned that a beam of light from the eye could instantaneously light up the stars every time someone blinked. Euclid stated the principle of shortest trajectory of light, and considered multiple reflections on flat and spherical mirrors. Ptolemy , in his treatise Optics , held an extramission-intromission theory of vision:
21798-489: The merits of Aristotelian and Euclidean ideas of optics, favouring the emission theory since it could better quantify optical phenomena. In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for lenses and curved mirrors . In the early 11th century, Alhazen (Ibn al-Haytham) wrote
21971-459: The mid-1980s when the comune had more than 2.8 million residents. After this, the population declined slowly as people began to move to nearby suburbs. Rome is in the Lazio region of central Italy on the Tiber ( Italian : Tevere ) river. The original settlement developed on hills that faced onto a ford beside the Tiber Island , the only natural ford of the river in this area. The Rome of
22144-565: The monumental centre, the palatine, and the largest baths, which continued to function until the Gothic siege of 537. The large baths of Constantine on the Quirinale were even repaired in 443, and the extent of the damage exaggerated and dramatised. However, the city gave an appearance overall of shabbiness and decay because of the large abandoned areas due to population decline. The population declined to 500,000 by 452 and 100,000 by 500 AD (perhaps larger, though no certain figure can be known). After
22317-660: The north Latium town of Sutri to the Church, starting its temporal power. In 756, Pepin the Short , after having defeated the Lombards, gave the Pope temporal jurisdiction over the Roman Duchy and the Exarchate of Ravenna , thus creating the Papal States . Since this period, three powers tried to rule the city: the pope, the nobility (together with the chiefs of militias, the judges, the Senate and
22490-405: The object and image distances are positive if the object and image are on opposite sides of the lens. Incoming parallel rays are focused by a converging lens onto a spot one focal length from the lens, on the far side of the lens. This is called the rear focal point of the lens. Rays from an object at a finite distance are focused further from the lens than the focal distance; the closer the object
22663-401: The optical explanations of astronomical phenomena such as lunar and solar eclipses and astronomical parallax . He was also able to correctly deduce the role of the retina as the actual organ that recorded images, finally being able to scientifically quantify the effects of different types of lenses that spectacle makers had been observing over the previous 300 years. After the invention of
22836-466: The outbreak of the Franco-Prussian War . Italian troops were able to capture Rome entering the city through a breach near Porta Pia . Pope Pius IX declared himself a prisoner in the Vatican . In 1871 the capital of Italy was moved from Florence to Rome. In 1870 the population of the city was 212,000, all of whom lived with the area circumscribed by the ancient city, and in 1920, the population
23009-676: The path taken between two points by a ray of light is the path that can be traversed in the least time. Geometric optics is often simplified by making the paraxial approximation , or "small angle approximation". The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing , which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications . Reflections can be divided into two types: specular reflection and diffuse reflection . Specular reflection describes
23182-504: The pope, the emperor, and each other. These were the times of Theodora and her daughter Marozia , concubines and mothers of several popes, and of Crescentius , a powerful feudal lord, who fought against the Emperors Otto II and Otto III . The scandals of this period forced the papacy to reform itself: the election of the pope was reserved to the cardinals, and reform of the clergy was attempted. The driving force behind this renewal
23355-453: The populace), and the Frankish king, as king of the Lombards, patricius, and Emperor. These three parties (theocratic, republican, and imperial) were a characteristic of Roman life during the entire Middle Ages. On Christmas night of 800, Charlemagne was crowned in Rome as Emperor by Pope Leo III : on that occasion, the city hosted for the first time the two powers whose struggle for control
23528-510: The problem of its quadrature . Although it appears that Torricelli reached his solution independently, the matter was still in dispute up to his death. Torricelli died of fever, most likely typhoid , in Florence on 25 October 1647, 10 days after his 39th birthday, and was buried at the Basilica of San Lorenzo . He left all his belongings to his adopted son Alessandro. "Belonging to that first period are his pamphlets on Solidi spherali, Contatti and
23701-511: The propagation of light in systems which cannot be solved analytically. Such models are computationally demanding and are normally only used to solve small-scale problems that require accuracy beyond that which can be achieved with analytical solutions. All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing . Gaussian beam propagation
23874-467: The publication of Galileo 's Dialogue Concerning the Two Chief World Systems , Torricelli wrote to Galileo of reading it "with the delight ... of one who, having already practiced all of geometry most diligently ... and having studied Ptolemy and seen almost everything of Tycho Brahe , Kepler and Longomontanus , finally, forced by the many congruences, came to adhere to Copernicus , and
24047-416: The rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarized much of Euclid and went on to describe a way to measure the angle of refraction , though he failed to notice the empirical relationship between it and
24220-423: The reflected light depending on the microscopic structure of the material. Many diffuse reflectors are described or can be approximated by Lambert's cosine law , which describes surfaces that have equal luminance when viewed from any angle. Glossy surfaces can give both specular and diffuse reflection. In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with
24393-415: The same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. The law also implies that mirror images are parity inverted, which we perceive as a left-right inversion. Images formed from reflection in two (or any even number of) mirrors are not parity inverted. Corner reflectors produce reflected rays that travel back in the direction from which
24566-407: The same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens. As with mirrors, upright images produced by a single lens are virtual, while inverted images are real. Lenses suffer from aberrations that distort images. Monochromatic aberrations occur because the geometry of the lens does not perfectly direct rays from each object point to
24739-478: The shore, where the south-western district of Ostia is located. The altitude of the central part of Rome ranges from 13 m (43 ft) above sea level (at the base of the Pantheon ) to 139 m (456 ft) above sea level (the peak of Monte Mario ). The Comune of Rome covers an overall area of about 1,285 km (496 sq mi), including many green areas. Public parks and nature reserves cover
24912-405: The simple equation 1 S 1 + 1 S 2 = 1 f , {\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}},} where S 1 is the distance from the object to the lens, θ 2 is the distance from the lens to the image, and f is the focal length of the lens. In the sign convention used here,
25085-464: The spectacle making centres in both the Netherlands and Germany. Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses rather than using the rudimentary optical theory of the day (theory which for the most part could not even adequately explain how spectacles worked). This practical development, mastery, and experimentation with lenses led directly to
25258-657: The successors of Peter, who is considered the first Bishop of Rome. The city thus became of increasing importance as the centre of the Catholic Church . After the Lombard invasion of Italy (569–572), the city remained nominally Byzantine, but in reality, the popes pursued a policy of equilibrium between the Byzantines , the Franks , and the Lombards . In 729, the Lombard king Liutprand donated
25431-444: The superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances. This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are in phase , both the wave crests and wave troughs align. This results in constructive interference and an increase in
25604-532: The support system from its control. Christianity in the form of the Nicene Creed became the official religion of the empire in 380, via the Edict of Thessalonica issued in the name of three emperors – Gratian, Valentinian II, and Theodosius I – with Theodosius clearly the driving force behind it. He was the last emperor of a unified empire: after his death in 395, his young children, Honorius and Arcadius , inherited
25777-569: The symbol of Rome's constituted republican government . Furthermore, Rome has been called Urbs Aeterna (The Eternal City), Caput Mundi (The Capital of the world ), Throne of St. Peter and Roma Capitale. While there have been discoveries of archaeological evidence of human occupation of the Rome area from approximately 14,000 years ago, the dense layer of much younger debris obscures Palaeolithic and Neolithic sites. Evidence of stone tools, pottery, and stone weapons attest to about 10,000 years of human presence. Several excavations support
25950-467: The telescope, Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope , using two convex lenses to produce higher magnification. Optical theory progressed in the mid-17th century with treatises written by philosopher René Descartes , which explained a variety of optical phenomena including reflection and refraction by assuming that light
26123-440: The term "light" is also often applied to infrared (0.7–300 μm) and ultraviolet radiation (10–400 nm). The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated. The existence of electromagnetic waves
26296-467: The three Macedonian Wars (212–168 BC) against Macedonia . The first Roman provinces were established at this time: Sicily , Sardinia and Corsica , Hispania , Macedonia , Achaea and Africa . From the beginning of the 2nd century BC, power was contested between two groups of aristocrats: the optimates , representing the conservative part of the Senate , and the populares , which relied on
26469-414: The three remaining months of his life. Torricelli's work led to first speculations about atmospheric pressure, and to the corollary invention of the mercury barometer (from the Greek word baros, meaning weight) -- the principle of which was described as early as 1631 by René Descartes , although there is no evidence that Descartes ever built such an instrument. The barometer arose from the need to solve
26642-631: The view that Rome grew from pastoral settlements on the Palatine Hill built above the area of the future Roman Forum . Between the end of the Bronze Age and the beginning of the Iron Age , each hill between the sea and the Capitoline Hill was topped by a village (on the Capitoline, a village is attested since the end of the 14th century BC). However, none of them yet had an urban quality. Nowadays, there
26815-433: The war as part of the " Italian economic miracle " of post-war reconstruction and modernisation in the 1950s and early 1960s. During this period, the years of la dolce vita ("the sweet life"), Rome became a fashionable city, with popular classic films such as Ben Hur , Quo Vadis , Roman Holiday and La Dolce Vita filmed in the city's iconic Cinecittà Studios . The rising trend in population growth continued until
26988-410: The water column below the piston but still pushes down on the surface of the water outside, thus causing the water to rise until its weight counterbalances the weight of the atmosphere. This hypothesis might have led him to a striking prediction: That a suction pump might only raise mercury, which is 13 times heavier than water, to 1/13 the height of the water column (76 centimeters) in a similar pump. (It
27161-424: Was 660,000. A significant portion lived outside the walls in the north and across the Tiber in the Vatican area. Soon after World War I in late 1922 Rome witnessed the rise of Italian Fascism led by Benito Mussolini , who led a march on the city . He did away with democracy by 1926, eventually declaring a new Italian Empire and allying Italy with Nazi Germany in 1938. Mussolini demolished fairly large parts of
27334-633: Was a Galileian in profession and sect". (The Vatican condemned Galileo in June 1633, and this was the only known occasion on which Torricelli openly declared himself to hold the Copernican view.) Aside from several letters, little is known of Torricelli's activities in the years between 1632 and 1641, when Castelli sent Torricelli's monograph of the path of projectiles to Galileo, then a prisoner in his villa at Arcetri . Although Galileo promptly invited Torricelli to visit, Torricelli did not accept until just three months before Galileo's death. The reason for this
27507-400: Was a famous instrument which used interference effects to accurately measure the speed of light. The appearance of thin films and coatings is directly affected by interference effects. Antireflective coatings use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with
27680-428: Was about 76 centimetres (30 in) above the surface of the mercury basin, producing a Torricellian vacuum above. This was also the first recorded incident of creating permanent vacuum. A second unambiguous prediction of Torricelli's sea of air hypothesis was made by Blaise Pascal , who argued, and proved, that the mercury column of the barometer should drop at higher elevations. Indeed, it dropped slightly on top of
27853-540: Was emitted by objects which produced it. This differed substantively from the ancient Greek emission theory. In the late 1660s and early 1670s, Isaac Newton expanded Descartes's ideas into a corpuscle theory of light , famously determining that white light was a mix of colours that can be separated into its component parts with a prism . In 1690, Christiaan Huygens proposed a wave theory for light based on suggestions that had been made by Robert Hooke in 1664. Hooke himself publicly criticised Newton's theories of light and
28026-649: Was first under the control of Odoacer and then became part of the Ostrogothic Kingdom before returning to East Roman control after the Gothic War , which devastated the city in 546 and 550 . Its population declined from more than a million in AD 210 to 500,000 in AD 273 to 35,000 after the Gothic War (535–554), reducing the sprawling city to groups of inhabited buildings interspersed among large areas of ruins, vegetation, vineyards and market gardens. It
28199-409: Was given a sound basic education. He then entered young Torricelli into a Jesuit College in 1624, possibly the one in Faenza itself, to study mathematics and philosophy until 1626, by which time his father, Gaspare, had died. The uncle then sent Torricelli to Rome to study science under the Benedictine monk Benedetto Castelli , professor of mathematics at the Collegio della Sapienza (now known as
28372-443: Was governed by creating regional dioceses. The existence of regional fiscal units from 286 served as the model for this unprecedented innovation. The emperor quickened the process of removing military command from governors. Henceforth, civilian administration and military command would be separate. He gave governors more fiscal duties and placed them in charge of the army logistical support system as an attempt to control it by removing
28545-408: Was heading south to fight the Hohenstaufen on behalf of the pope, was appointed Senator. Charles founded the Sapienza , the university of Rome. In that period the pope died, and the cardinals, summoned in Viterbo , could not agree on his successor. This angered the people of the city, who then unroofed the building where they met and imprisoned them until they had nominated the new pope; this marked
28718-444: Was known as the caput Mundi , i.e. the capital of the known world, an expression which had already been used in the Republican period. During its first two centuries, the empire was ruled by emperors of the Julio-Claudian , Flavian (who built an eponymous amphitheatre known as the Colosseum ), and Antonine dynasties. This time was also characterised by the spread of the Christian religion, preached by Jesus Christ in Judea in
28891-467: Was predicted in 1865 by Maxwell's equations . These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves. Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered. Many simplified approximations are available for analysing and designing optical systems. Most of these use
29064-421: Was published in 1644. Little was known about Torricelli in regard to his works in geometry when he accepted the honorable position, but after he published Opera Geometrica two years later, he became highly esteemed in that discipline. "He was interested in Optics , and invented a method whereby microscopic lenses might be made of glass which could be easily melted in a lamp." As a result, he designed and built
29237-405: Was restored in part much later). In 330, he transformed Byzantium into Constantinople , which became his new capital. However, it was not officially anything more than an imperial residence like Milan , Trier or Nicomedia until given a city prefect in 359 by Constantius II . Constantine, following Diocletian's reforms. regionalised the administration, which fundamentally changed the way it
29410-440: Was that Torricelli's mother, Caterina Angetti died. "(T)his short intercourse with the great mathematician enabled Torricelli to finish the fifth dialogue under the personal direction of its author; it was published by Viviani, another pupil of Galileo, in 1674." After Galileo's death on 8 January 1642, Grand Duke Ferdinando II de' Medici asked Torricelli to succeed Galileo as the grand-ducal mathematician and chair of mathematics at
29583-399: Was the monk Ildebrando da Soana , who once elected pope under the name of Gregory VII became involved into the Investiture Controversy against Emperor Henry IV . Subsequently, Rome was sacked and burned by the Normans under Robert Guiscard who had entered the city in support of the Pope, then besieged in Castel Sant'Angelo . During this period, the city was autonomously ruled by
29756-409: Was to be a constant of the Middle Ages. This event marks the beginning of the Carolingian Empire , the first phase of the Holy Roman Empire . In 846, Muslim Arabs unsuccessfully stormed the city's walls , but managed to loot St. Peter 's and St. Paul's basilica, both outside the city wall. After the decay of Carolingian power , Rome fell prey to feudal chaos: several noble families fought against
29929-431: Was united as the Kingdom of Italy in 1861 with the temporary capital in Florence . That year Rome was declared the capital of Italy even though it was still under the Pope's control. During the 1860s, the last vestiges of the Papal States were under French protection thanks to the foreign policy of Napoleon III . French troops were stationed in the region under Papal control. In 1870 the French troops were withdrawn due to
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