Bonaventura Francesco Cavalieri was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infinitesimal calculus, the introduction of logarithms to Italy. Cavalieri's principle in geometry anticipated integral calculus. Born in Milan, Cavalieri joined the Jesuates order at the age of fifteen and remained a member until his death, he studied theology in the monastery of San Gerolamo in Milan, geometry at the University of Pisa. He published eleven books, his first being published in 1632, he worked on the problems of optics and motion. His astronomical and astrological work remained marginal to these main interests, though his last book, Trattato della ruota planetaria perpetua, was dedicated to the former, he was introduced to Galileo Galilei through ecclesiastical contacts. Galileo exerted a strong influence on Cavalieri encouraging him to work on his new method and suggesting fruitful ideas, Cavalieri would write at least 112 letters to Galileo.
Galileo said of Cavalieri, "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." He benefited from the patronage of Cesare Marsili. Cavalieri's first book was Lo Specchio Ustorio, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections. In this book he developed the theory of mirrors shaped into parabolas and ellipses, various combinations of these mirrors; the work was purely theoretical since the needed mirrors could not be constructed with the technologies of the time, a limitation well understood by Cavalieri. Inspired by earlier work by Galileo, Cavalieri developed a new geometrical approach called the method of indivisibles to calculus and published a treatise on the topic, Geometria indivisibilibus continuorum nova quadam ratione promota. In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas; such elements are called indivisibles of area and volume and provide the building blocks of Cavalieri's method.
As an application, he computed the areas under the curves y = x n – an early integral –, known as Cavalieri's quadrature formula. Cavalieri is known for Cavalieri's principle, which states that the volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. Two cross-sections correspond if they are intersections of the body with planes equidistant from a chosen base plane. Cavalieri constructed a hydraulic pump for his monastery and published tables of logarithms, emphasizing their practical use in the fields of astronomy and geography, he died in Bologna. According to Gilles-Gaston Granger, Cavalieri belongs with Newton, Pascal and MacLaurin as one of those who in the 17th and 18th centuries "redefine the mathematical object"; the lunar crater Cavalerius is named for Cavalieri. Evangelista Torricelli Stefano degli Angeli Cavalieri's quadrature formula The Galileo Project: Cavalieri Elogj di Galileo Galilei e di Bonaventura Cavalieri by Giuseppe Galeazzi, Milan, 1778 Bonaventura Cavalieri by Antonio Favaro, vol. 31 of Amici e corrispondenti di Galileo Galilei, C.
Ferrari, 1915. Online texts by Cavalieri: Lo specchio ustorio: overo, Trattato delle settioni coniche... Directorium generale uranometricum Geometria Indivisibilibus Sfera astronomica Biographies: O'Connor, John J.. Short biography on bookrags.com Fabroni, Angelo. "Bonaventura Cavalerius". Vitae Italorum doctrina excellentium qui saeculis XVII. Et XVIII. Floruerunt. Pisa. I: 262–301. Modern mathematical or historical research: Infinitesimal Calculus On its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. More information about the method of Cavalieri Cavalieri Integration
The public domain consists of all the creative works to which no exclusive intellectual property rights apply. Those rights may have been forfeited, expressly waived, or may be inapplicable; the works of William Shakespeare and Beethoven, most early silent films, are in the public domain either by virtue of their having been created before copyright existed, or by their copyright term having expired. Some works are not covered by copyright, are therefore in the public domain—among them the formulae of Newtonian physics, cooking recipes, all computer software created prior to 1974. Other works are dedicated by their authors to the public domain; the term public domain is not applied to situations where the creator of a work retains residual rights, in which case use of the work is referred to as "under license" or "with permission". As rights vary by country and jurisdiction, a work may be subject to rights in one country and be in the public domain in another; some rights depend on registrations on a country-by-country basis, the absence of registration in a particular country, if required, gives rise to public-domain status for a work in that country.
The term public domain may be interchangeably used with other imprecise or undefined terms such as the "public sphere" or "commons", including concepts such as the "commons of the mind", the "intellectual commons", the "information commons". Although the term "domain" did not come into use until the mid-18th century, the concept "can be traced back to the ancient Roman Law, as a preset system included in the property right system." The Romans had a large proprietary rights system where they defined "many things that cannot be owned" as res nullius, res communes, res publicae and res universitatis. The term res nullius was defined as things not yet appropriated; the term res communes was defined as "things that could be enjoyed by mankind, such as air and ocean." The term res publicae referred to things that were shared by all citizens, the term res universitatis meant things that were owned by the municipalities of Rome. When looking at it from a historical perspective, one could say the construction of the idea of "public domain" sprouted from the concepts of res communes, res publicae, res universitatis in early Roman law.
When the first early copyright law was first established in Britain with the Statute of Anne in 1710, public domain did not appear. However, similar concepts were developed by French jurists in the 18th century. Instead of "public domain", they used terms such as publici juris or propriété publique to describe works that were not covered by copyright law; the phrase "fall in the public domain" can be traced to mid-19th century France to describe the end of copyright term. The French poet Alfred de Vigny equated the expiration of copyright with a work falling "into the sink hole of public domain" and if the public domain receives any attention from intellectual property lawyers it is still treated as little more than that, left when intellectual property rights, such as copyright and trademarks, expire or are abandoned. In this historical context Paul Torremans describes copyright as a, "little coral reef of private right jutting up from the ocean of the public domain." Copyright law differs by country, the American legal scholar Pamela Samuelson has described the public domain as being "different sizes at different times in different countries".
Definitions of the boundaries of the public domain in relation to copyright, or intellectual property more regard the public domain as a negative space. According to James Boyle this definition underlines common usage of the term public domain and equates the public domain to public property and works in copyright to private property. However, the usage of the term public domain can be more granular, including for example uses of works in copyright permitted by copyright exceptions; such a definition regards work in copyright as private property subject to fair-use rights and limitation on ownership. A conceptual definition comes from Lange, who focused on what the public domain should be: "it should be a place of sanctuary for individual creative expression, a sanctuary conferring affirmative protection against the forces of private appropriation that threatened such expression". Patterson and Lindberg described the public domain not as a "territory", but rather as a concept: "here are certain materials – the air we breathe, rain, life, thoughts, ideas, numbers – not subject to private ownership.
The materials that compose our cultural heritage must be free for all living to use no less than matter necessary for biological survival." The term public domain may be interchangeably used with other imprecise or undefined terms such as the "public sphere" or "commons", including concepts such as the "commons of the mind", the "intellectual commons", the "information commons". A public-domain book is a book with no copyright, a book, created without a license, or a book where its copyrights expired or have been forfeited. In most countries the term of protection of copyright lasts until January first, 70 years after the death of the latest living author; the longest copyright term is in Mexico, which has life plus 100 years for all deaths since July 1928. A notable exception is the United States, where every book and tale published prior to 1924 is in the public domain.
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length the two regions have equal areas. 3-dimensional case: Suppose two regions in three-space are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards integral calculus, while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits but did not use infinitesimals. Cavalieri's principle was called the method of indivisibles, the name it was known by in Renaissance Europe.
Cavalieri developed a complete theory of indivisibles, elaborated in his Geometria indivisibilibus continuorum nova quadam ratione promota and his Exercitationes geometricae sex. In the 3rd century BC, using a method resembling Cavalieri's principle, was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems. In the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume; the transition from Cavalieri's indivisibles to Evangelista Torricelli's and John Wallis's infinitesimals was a major advance in the history of the calculus. The indivisibles were entities of codimension 1, so that a plane figure was thought as made out of an infinity of 1-dimensional lines. Meanwhile, infinitesimals were entities of the same dimension as the figure. Applying the formula for the sum of an arithmetic progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞.
If one knows that the volume of a cone is 1 3 one can use Cavalieri's principle to derive the fact that the volume of a sphere is 4 3 π r 3, where r is the radius. That is done as follows: Consider a sphere of radius r and a cylinder of radius r and height r. Within the cylinder is the cone whose apex is at the center of one base of the cylinder and whose base is the other base of the cylinder. By the Pythagorean theorem, the plane located y units above the "equator" intersects the sphere in a circle of area π; the area of the plane's intersection with the part of the cylinder, outside of the cone is π. As we can see, the area of every intersection of the circle with the horizontal plane located at any height y equals the area of the intersection of the plane with the part of the cylinder, "outside" of the cone; the aforementioned volume of the cone is 1 3 of the volume of the cylinder, thus the volume outside of the cone is 2 3 the volume of the cylinder. Therefore the volume of the upper half of the sphere is 2 3 of the volume of the cylinder.
The volume of the cylinder is base × height = π r 2 ⋅ r = π r 3 Therefore the volume of the upper half-sphere is 2 3 π r 3 and that of the whole sphere is 4 3 π r 3. The fact that the volume of any pyramid, regardless of the shape of the base, whether circular as in the case of a cone, or square as in the case of the Egyptian pyramids, or any other shape, is × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case. One may establish it in a single case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of those three volumes by means of Cavalieri's principle. In fact, Cavalieri's principle or similar infinitesimal argument is necessary to compute the volume of cones and pyramids
Society of Jesus
The Society of Jesus is a scholarly religious congregation of the Catholic Church for men founded by Ignatius of Loyola and approved by Pope Paul III. The members are called Jesuits; the society is engaged in evangelization and apostolic ministry in 112 nations. Jesuits work in education, intellectual research, cultural pursuits. Jesuits give retreats, minister in hospitals and parishes, sponsor direct social ministries, promote ecumenical dialogue. Saint Ignatius of Loyola, a Basque nobleman from the Pyrenees area of northern Spain, founded the society after discerning his spiritual vocation while recovering from a wound sustained in the Battle of Pamplona, he composed the Spiritual Exercises to help others follow the teachings of Jesus Christ. In 1534, Ignatius and six other young men, including Francis Xavier and Peter Faber and professed vows of poverty and obedience, including a special vow of obedience to the Pope in matters of mission direction and assignment. Ignatius's plan of the order's organization was approved by Pope Paul III in 1540 by a bull containing the "Formula of the Institute".
Ignatius was a nobleman who had a military background, the members of the society were supposed to accept orders anywhere in the world, where they might be required to live in extreme conditions. Accordingly, the opening lines of the founding document declared that the society was founded for "whoever desires to serve as a soldier of God to strive for the defence and propagation of the faith and for the progress of souls in Christian life and doctrine." Jesuits are thus sometimes referred to colloquially as "God's soldiers", "God's marines", or "the Company", which evolved from references to Ignatius' history as a soldier and the society's commitment to accepting orders anywhere and to endure any conditions. The society participated in the Counter-Reformation and in the implementation of the Second Vatican Council; the Society of Jesus is consecrated under the patronage of Madonna Della Strada, a title of the Blessed Virgin Mary, it is led by a Superior General. The headquarters of the society, its General Curia, is in Rome.
The historic curia of Ignatius is now part of the Collegio del Gesù attached to the Church of the Gesù, the Jesuit mother church. In 2013, Jorge Mario Bergoglio became the first Jesuit to be elected Pope, taking the name Pope Francis; as of 2012, the Jesuits formed the largest single religious order of priests and brothers in the Catholic Church. The Jesuits have experienced a decline in numbers in recent decades; as of 2017 the society had 16,088 members, 11,583 priests and 4,505 Jesuits in formation, which includes brothers and scholastics. This represents a 42.6 percent decline since 1977, when the society had a total membership of 28,038, of which 20,205 were priests. This decline is most pronounced in Europe and the Americas, with modest membership gains occurring in Asia and Africa. There seems to be no "Pope Francis effect" in counteracting the fall of vocations among the Jesuits; the society is divided into 83 provinces along with six independent regions and ten dependent regions. On 1 January 2007, members served in 112 nations on six continents with the largest number in India and the US.
Their average age was 57.3 years: 63.4 years for priests, 29.9 years for scholastics, 65.5 years for brothers. The current Superior General of the Jesuits is Arturo Sosa; the society is characterized by its ministries in the fields of missionary work, human rights, social justice and, most notably, higher education. It operates colleges and universities in various countries around the world and is active in the Philippines and India. In the United States the Jesuits have historical ties to 28 colleges and universities and 61 high schools; the degree to which the Jesuits are involved in the administration of each institution varies. As of September 2018, 15 of the 28 Jesuit universities in the US had non-Jesuit lay presidents. According to a 2014 article in The Atlantic, "the number of Jesuit priests who are active in everyday operations at the schools isn’t nearly as high as it once was". Worldwide it runs 172 colleges and universities. A typical conception of the mission of a Jesuit school will contain such concepts as proposing Christ as the model of human life, the pursuit of excellence in teaching and learning, lifelong spiritual and intellectual growth, training men and women for others.
Ignatius laid out his original vision for the new order in the "Formula of the Institute of the Society of Jesus", "the fundamental charter of the order, of which all subsequent official documents were elaborations and to which they had to conform." He ensured that his formula was contained in two papal bulls signed by Pope Paul III in 1540 and by Pope Julius III in 1550. The formula expressed the nature, community life, apostolate of the new religious order, its famous opening statement echoed Ignatius' military background: Whoever desires to serve as a soldier of God beneath the banner of the Cross in our Society, which we desire to be designated by the Name of Jesus, to serve the Lord alone and the Church, his spouse, under the Roman Pontiff, the Vicar of Christ on earth, after a solemn vow of perpetual chastity and obedience, keep what follows in mind. He is a member of a Society founded chiefly for this purpose: to strive for the defence and propagation of the faith and for the progress of souls in Christian life and doctrine, by means of public preaching and any other ministration whatsoever of the Word of God, further by means of ret
The Benedictines the Order of Saint Benedict, are a monastic Catholic religious order of monks and nuns that follow the Rule of Saint Benedict. They are sometimes called the Black Monks, in reference to the colour of the members' religious habits. Despite being called an order, the Benedictines do not operate under a single hierarchy but are instead organised as a collection of independent monastic communities, with each community within the order maintaining its own autonomy. Unlike other religious orders, the Benedictines do not have a superior general or motherhouse with universal jurisdiction. Instead, the order is represented internationally by the Benedictine Confederation, an organisation, set up in 1893 to represent the order's shared interests; the monastery at Subiaco in Italy, established by Saint Benedict of Nursia c. 529, was the first of the dozen monasteries he founded. He founded the Abbey of Monte Cassino. There is no evidence, that he intended to found an order and the Rule of Saint Benedict presupposes the autonomy of each community.
When Monte Cassino was sacked by the Lombards about the year 580, the monks fled to Rome, it seems probable that this constituted an important factor in the diffusion of a knowledge of Benedictine monasticism. It was from the monastery of St. Andrew in Rome that Augustine, the prior, his forty companions set forth in 595 on their mission for the evangelization of England. At various stopping places during the journey, the monks left behind them traditions concerning their rule and form of life, also some copies of the Rule. Lérins Abbey, for instance, founded by Honoratus in 375 received its first knowledge of the Benedictine Rule from the visit of St. Augustine and his companions in 596. Gregory of Tours says that at Ainay Abbey, in the sixth century, the monks "followed the rules of Basil, Cassian and other fathers and using whatever seemed proper to the conditions of time and place", doubtless the same liberty was taken with the Benedictine Rule when it reached them. In Gaul and Switzerland, it supplemented the much stricter Irish or Celtic Rule introduced by Columbanus and others.
In many monasteries it entirely displaced the earlier codes. By the ninth century, the Benedictine had become the standard form of monastic life throughout the whole of Western Europe, excepting Scotland and Ireland, where the Celtic observance still prevailed for another century or two. Through the work of Benedict of Aniane, it became the rule of choice for monasteries throughout the Carolingian empire. Monastic scriptoria flourished from the ninth through the twelfth centuries. Sacred Scripture was always at the heart of every monastic scriptorium; as a general rule those of the monks who possessed skill as writers made this their chief, if not their sole active work. An anonymous writer of the ninth or tenth century speaks of six hours a day as the usual task of a scribe, which would absorb all the time available for active work in the day of a medieval monk. In the Middle Ages monasteries were founded by the nobility. Cluny Abbey was founded by William I, Duke of Aquitaine in 910; the abbey was noted for its strict adherence to the Rule of St. Benedict.
The abbot of Cluny was the superior of all the daughter houses, through appointed priors. One of the earliest reforms of Benedictine practice was that initiated in 980 by Romuald, who founded the Camaldolese community; the dominance of the Benedictine monastic way of life began to decline towards the end of the twelfth century, which saw the rise of the Franciscans and Dominicans. Benedictines took a fourth vow of "stability". Not being bound by location, the mendicants were better able to respond to an "urban" environment; this decline was further exacerbated by the practice of appointing a commendatory abbot, a lay person, appointed by a noble to oversee and to protect the goods of the monastery. Oftentimes, this resulted in the appropriation of the assets of monasteries at the expense of the community which they were intended to support; the English Benedictine Congregation is the oldest of the nineteen Benedictine congregations. Augustine of Canterbury and his monks established the first English Benedictine monastery at Canterbury soon after their arrival in 597.
Other foundations followed. Through the influence of Wilfrid, Benedict Biscop, Dunstan, the Benedictine Rule spread with extraordinary rapidity, in the North it was adopted in most of the monasteries, founded by the Celtic missionaries from Iona. Many of the episcopal sees of England were founded and governed by the Benedictines, no fewer than nine of the old cathedrals were served by the black monks of the priories attached to them. Monasteries served as places of refuge for the weak and homeless; the monks studied the healing properties of plants and minerals to alleviate the sufferings of the sick. Germany was evangelized by English Benedictines. Willibrord and Boniface preached there in the seventh and eighth centuries and founded several abbeys. In the English Reformation, all monasteries were dissolved and their lands confiscated by the Crown, forcing their Catholic members to flee into exile on the Continent. During the 19th century they were able to return to England, including to Selby Abbey in Yorkshire, one of the few great monastic churches to survive the Dissolution.
St. Mildred's Priory, on the Isle of Thanet, was built in 1027 on the site of an abbey founded in 670 by the daughter of the first Christian King of Kent; the priory is home to a community of Benedictine nuns. Five of
Avignon is a commune in south-eastern France in the department of Vaucluse on the left bank of the Rhône river. Of the 90,194 inhabitants of the city, about 12,000 live in the ancient town centre enclosed by its medieval ramparts. Between 1309 and 1377, during the Avignon Papacy, seven successive popes resided in Avignon and in 1348 Pope Clement VI bought the town from Joanna I of Naples. Papal control persisted until 1791; the town is now the capital of the Vaucluse department and one of the few French cities to have preserved its ramparts. The historic centre, which includes the Palais des Papes, the cathedral, the Pont d'Avignon, became a UNESCO World Heritage Site in 1995; the medieval monuments and the annual Festival d'Avignon have helped to make the town a major centre for tourism. The earliest forms of the name were reported by the Greeks: Аὐενιὼν = Auenion Άουεννίων = Aouennion; the Roman name Avennĭo Cavarum, i.e. "Avignon of Cavares" shows that Avignon was one of the three cities of the Celtic-Ligurian tribe of Cavares, along with Cavaillon and Orange.
The current name dates to a pre-Indo-European or pre-Latin theme ab-ên with the suffix -i-ōn This theme would be a hydronym – i.e. a name linked to the river, but also an oronym of terrain. The Auenion of the 1st century BC was Latinized to Avennĭo, -ōnis in the 1st century and was written Avinhon in classic Occitan spelling or Avignoun in Mistralian spelling The inhabitants of the commune are called avinhonencs or avignounen in both Occitan and Provençal dialect. Avignon is on the left bank of the Rhône river, a few kilometres above its confluence with the Durance, about 580 km south-east of Paris, 229 km south of Lyon and 85 km north-north-west of Marseille. On the west it shares a border with the department of Gard and the communes of Villeneuve-lès-Avignon and Les Angles and to the south it borders the department of Bouches-du-Rhône and the communes of Barbentane, Rognonas, Châteaurenard, Noves; the city is in the vicinity of Orange, Nîmes, Arles, Salon-de-Provence, Marseille. Directly contiguous to the east and north are the communes of Caumont-sur-Durance, Morières-lès-Avignon, Le Pontet, Sorgues.
The region around Avignon is rich in limestone, used for building material. For example, the current ramparts, measuring 4,330 metres long, were built with the soft limestone abundant in the region called mollasse burdigalienne. Enclosed by the ramparts, the Rocher des Doms is a limestone elevation of urgonian type, 35 metres high and is the original core of the city. Several limestone massifs are present around the commune and they are the result of the oceanisation of the Ligurian-Provençal basin following the migration of the Sardo-Corsican block; the other significant elevation in the commune is the Montfavet Hill – a wooded hill in the east of the commune. The Rhone Valley is an old alluvial zone: loose deposits cover much of the ground, it consists of sandy alluvium more or less coloured with pebbles consisting of siliceous rocks. The islands in the Rhone, such as the Île de la Barthelasse, were created by the accumulation of alluvial deposits and by the work of man; the relief is quite low despite the creation of mounds allowing local protection from flooding.
In the land around the city there are clay, silt and limestone present. The Rhone passes the western edge of the city but is divided into two branches: the Petit Rhône, or "dead arm", for the part that passes next to Avignon and the Grand Rhône, or "live arm", for the western channel which passes Villeneuve-lès-Avignon in the Gard department; the two branches are separated by the Île de la Barthelasse. The southernmost tip of the Île de la Barthelasse once formed of a separated island, the L'Île de Piot; the banks of the Rhone and the Île de la Barthelasse are subject to flooding during autumn and March. The publication Floods in France since the 6th century until today – research and documentation by Maurice Champion tells about a number of them, they have never stopped as shown by the floods in 1943–1944 and again on 23 January 1955 and remain important today – such as the floods of 2 December 2003. As a result, a new risk mapping has been developed; the Durance flows along the southern boundary of the commune into the Rhone and marks the departmental boundary with Bouches-du-Rhône.
It is a river, considered "capricious" and once feared for its floods (it was once called the "3rd scourge of Provence" as well as for its low water: the Durance has both Alpine and Mediterranean morphology, unusual. There are many natural and artificial water lakes in the commune such as the Lake of Saint-Chamand east of the city. There have been many diversions throughout the course of history, such as feeding the moat surrounding Avignon or irrigating crops. In the 10th century part of the waters from the Sorgue d'Entraigues were diverted and today pass under the ramparts to enter the city.. This watercourse is called the Vaucluse Canal but Avignon people still call it the Sorgue or Sorguette, it is visible in the city in the famous Rue des teinturiers. It fed the moat around the first ramparts fed the moat on the newer east