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Montpelier, Vermont

Montpelier  is the capital of the U.S. state of Vermont and the county seat of Washington County. The site of Vermont's state government, it is the least populated state capital in the United States. As of the 2020 census, the population was 8,074, with a daytime population growth of about 21,000 due to the large number of jobs within city limits. The Vermont College of Fine Arts is located in the municipality. It was named after Montpellier, a city in the south of France.

Montpelier was chartered as a town by proprietors from Massachusetts and western Vermont on August 14, 1781, and the Town of Montpelier was granted municipal powers by the "Governor, Council and General Assembly of the Freemen of the State of Vermont". The first permanent settlement began in May 1787, and a town meeting was established in 1791. The city received a French name because the Franco-American alliance during the Revolutionary War had sparked widespread Francophilia. Montpelier was selected as state capital in 1805, and citizens of the town donated funds to build the first state house. The legislature chartered the City of Montpelier in 1894, and it was organized at a town meeting the next year.

Montpelier is in the north-central area of the state. The Winooski River flows by the south side of the city, passing by the state house and continuing northwest into Chittenden County. The North Branch River, a tributary of the Winooski, also flows through the city. Flooding is a periodic problem in Montpelier, with three serious floods occurring in 1927, 1992, and 2023. In 1875, fires destroyed 38 buildings in the city, and the commercial area was rebuilt with brick to avoid a repeat of the disaster. The state government contributes to the city's economy, which also relies on insurance and tourism. The insurance industry in Montpelier dates back to the early 19th century. Historically, tradesmen shops and mills along the North Branch also fueled the economy, and sheds for finishing granite from Barre operated along the Winooski. Ski areas are also located near the city.

Travelers can reach the city via the Amtrak Montpelier station, which is located 2 miles west of the downtown area and offers regular service on the Vermonter train. Private planes can also use the nearby Edward F. Knapp State Airport in Berlin, Vermont, but the closest commercial air service is via the Patrick Leahy Burlington International Airport 35 mi to the northwest.

History

The meadows and flats of the Winooski River were well known among natives for their corn-raising capacities. The natural site of Montpelier made it a favorite residence for the natives who first inhabited the land. The level plain of nearly two hundred acres of the rich farmland, sheltered from winds by the surrounding valley made the area comparatively warm and comfortable. Its position near the confluence of many streams allowed for favorable hunting, fishing, and trading. Native mounds, tombstones, and other memorials of aboriginal life and death were found in Montpelier, or in the vicinity, by the first settlers, and traces of some of them still remain.

Between 1600 and 1800, European settlers began to arrive in the region. Soon after, war, disease, and dispersal virtually destroyed the Native American settlements. However, evidence suggests some Native Americans remained in the area as late as the mid-1800s.

Originally chartered on August 14, 1781, the Town of Montpelier was granted municipal powers by the "Governor, Council and General Assembly of the Freemen of the State of Vermont". The first permanent settlement began in May 1787, when Colonel Jacob Davis and General Parley Davis arrived from Charlton, Massachusetts. General Davis surveyed the land, while Colonel Davis cleared forest and erected a large log house on the west side of the North Branch of the Winooski River. His family moved in the following winter.

Colonel Davis selected the name "Montpelier" after the French city of Montpellier, capital of the department of Hérault. There was a general enthusiasm for things French as a result of the country's aid to the American colonies during the Revolutionary War. The settlement grew quickly, and by 1791 the population reached 117.

The configuration of the early village was strongly influenced by geography. As early as 1799 a bridge was constructed across the Winooski River to Berlin.

The Town's Charter was reissued on February 6, 1804, to include a boundary description of the lands granted to the Town's inhabitants and proprietors. The confluence of the Winooski, North Branch and Dog Rivers provided a central point for the local population and commerce.

By 1805, the town had a population of 1,200. In that year, the state legislature sought a permanent home. Montpelier was selected because of its central location and accessibility, and because local residents provided land and money. A humble State House was soon constructed on State Street.

In 1825, the Marquis de Lafayette visited Montpelier on a triumphal tour of the United States, 50 years after the Revolutionary War.

The town developed into a center for manufacturing, especially after the Central Vermont Railway opened in Montpelier on June 20, 1849. In response to Montpelier's growth and changing demographics, on November 9, 1848, the General Assembly divided the original Town into two district municipal corporations. The towns of East Montpelier and Montpelier were created. Later, in an attempt to modernize its form of government, the town was reconstituted as the Village of Montpelier.

By 1858, the layout of the main streets paralleling the rivers was in place. The downtown street pattern has changed little since that time.

Ten thousand people turned out to greet Major General Philip Sheridan in 1867 when he visited to address the fourth annual meeting of Vermont former Union officers. He particularly thanked Vermont veterans of the Civil War for their performance at the Battle of Cedar Creek.

In 1875, a large fire destroyed many downtown buildings.

The village had the first municipal water driven hydro system in Vermont in 1884. Water pressure generated sufficient electricity for streetlights.

The first charter of Montpelier was granted in 1894 and was amended shortly thereafter in 1898, and again in 1900 and 1912. The first amendment permitted the city to annex a part of the Town of Berlin; the latter enactments amended the 1898 charter to deal with such matters as water works, the relationship between the city and the Washington County Grammar School, and composition of the City Council.

The state proclaimed October 12, 1899, as "Dewey Day" to honor native son George Dewey, the hero of Battle of Manila Bay in the Spanish–American War. Thousands turned out from the state to his hometown of Montpelier for the celebration. In 1899, Hubbard Park was established with a donation of land, known as "Hubbard Hill", bequeathed to the City of Montpelier by John Erastus Hubbard (1847–1899) with the intent to "preserve wilderness" for future generations. In 1911, additional land was donated and from 1915 to 1930 an observation tower was constructed on this donated land.

In 1927, after a particularly wet summer and fall, heavy rains began on the evening of November 2 that continued until the morning of November 4. The heaviest rain fell on November 3, when more than seven inches fell in a six-hour period. The prolonged heavy rains on top of the already saturated soil from the summer and fall proved to be more than the watercourses could handle. Brooks and rivers overflowed, carrying trees and logs in their wake. Dams, bridges and embankments were destroyed. Buildings were submerged, farm animals drowned, and homes and barns were swept away. Rivers reached 13 feet or more above their normal depths. Flood waters gradually receded, leaving behind silt, gravel and debris. At least a foot of mud was left on the floors of downtown stores.

At the time, only two stores in Montpelier carried flood insurance. The staggering loss represented an average of $400 for every man, woman, and child in town—equivalent to roughly $5,760 in 2018 dollars. In the days following the flood, Vermont was widely praised for its recovery efforts. President Calvin Coolidge, in particular, hailed the “indomitable spirit” of Vermonters, of whom he was one.

In response to the damage suffered by Montpelier and surrounding communities in the Great Flood of 1927, the Civilian Conservation Corps built the Wrightsville Dam during a period from 1933 to 1935. The resulting reservoir, Wrightsville Reservoir, required the disbandment and flooding of the village of Wrightsville, which contained at least 30 built structures at the time.

The City of Montpelier grew slowly in the late 19th and early 20th centuries during the period of intensive out-migration from the state to new lands in the West, or to industrial centers elsewhere in New England. Montpelier was already established as a government, market, service and industrial center in the region. When the automobile arrived, new state highways were routed to the city limits, and traffic then circulated through the original streets of the city. In 1954, a new bridge was constructed at Bailey Avenue which linked to an extension of Winooski Avenue, now Memorial Drive, and diverted some of the traffic from the downtown area.

An early spring thaw in March 1992 caused an ice jam to form in the Winooski River downstream of the Bailey Avenue bridge in Montpelier. In less than an hour, water levels in the Winooski and North Branch rivers rose upstream of the ice jam and flooded downtown Montpelier. The damage shut down 120 businesses, left 50 residents without homes, disrupted the operations of state government, and caused upwards of $5 million in damage.

Heavy rains on July 10-11, 2023, resulted in catastrophic flooding in Montpelier and many other parts of Vermont. In addition to many residences, the waters severely affected multiple downtown businesses as well as the city's fire department, police department, and City Hall.

Geography
Montpelier is located in the north-central area of Vermont. The city center is a flat clay zone (elevation 520 ft), surrounded by hills and granite ledges. Towne Hill runs in a 2 mi ridge (~900 ft) along the northern edge of the city.

According to the United States Census Bureau, the city has a total area of 10.3 sqmi, of which 10.2 sqmi is land and 0.10% is water. The Winooski River flows west along the south edge of downtown village and is fed by several smaller tributaries that cut through residential districts. Montpelier has been subject to periodic flooding in the flat city center, with two major floods occurring in 1927 and in 1992.

On its borders are the towns of Middlesex to the west, Berlin to the south, and East Montpelier to the north and east. Montpelier lies near the geographic center of the state. Though it does not share a border, Montpelier is frequently associated with the nearby city of Barre, and the two are often referred to together as "Barre-Montpelier".

Climate
Montpelier features a humid continental climate (Köppen Dfb), with long, cold, and snowy winters, short springs and autumns, and warm, humid summers. From January to July, daily means range from 16.4 to 67.3 F. In winter, lows fall below 0 F on 24 mornings and daytime highs stay below freezing for the majority of afternoons from December to February. Snow is also frequent and remains on the ground for long stretches throughout the winter, though thaws are by no means infrequent. Average annual snowfall is 94.2 in. Summers are warm and often humid, with 2 or 3 days above 90 F, but rarely reaching 95 F.

Extremes have ranged from -34 F in January 1981 to 97 F, most recently recorded in July 1977.

Demographics
Along with Barre, the city forms a small micropolitan area in the center of the state; together they are known as the twin cities.

At the 2019 American Community Survey, an estimated 7,852 people lived in the city. At the census of 2010, there were 7,855 people, 3,739 households, and 1,940 families residing in the city. The population density was 784.0 /mi2. There were 3,899 housing units at an average density of 380.4 /mi2.

At the 2010 census, there were 3,739 households, out of which 26.0% had children under the age of 18 years living with them, 38.5% were married couples living together, 10.1% had a female householder with no husband present, and 48.1% were non-families. 39.4% of all households were made up of individuals, and 13.1% had someone living alone who was 65 years of age or older. The average household size was 2.09 and the average family size was 2.84.

In the city, the population was spread out, with 21.3% under the age of 18, 8.6% from 18 to 24, 28.2% from 25 to 44, 27.1% from 45 to 64, and 14.9% who were 65 years of age or older. The median age was 40 years. For every 100 females, there were 84.2 males. For every 100 females age 18 and over, there were 82.0 males.

The median income for a household in the city was $37,513, and the median income for a family was $51,818. Males had a median income of $35,957 versus $29,442 for females. The per capita income for the city was $22,599. About 7.2% of families and 9.8% of the population were below the poverty line, including 12.9% of those under age 18 and 5.7% of those age 65 or over.

In 2019, there were 3,668 households with an average of 2.03 persons per household. The city had an owner-occupied housing rate of 55.6% and median value of owner-occupied housing rate of $252,600. The median monthly cost of an owner-occupied housing unit was $1,827 and without a mortgage $822. The city had a median gross rent of $1,022.

Race and ethnicity
In 2019, the racial and ethnic makeup of Montpelier was 92.2% non-Hispanic white, 1.6% Black or African American, 1.5% Asian, 2.4% two or more races, and 2.3% Hispanic or Latin American of any race. Among the population in 2019, 4.1% were foreign-born persons. In 2010, the racial makeup of the city was 93.7% White, 1.0% African American, 0.3% Native American, 2.2% Asian, 0.01% Pacific Islander, 0.39% from other races, and 2.2% from two or more races. Hispanic or Latino of any race were 2.1% of the population.

Economy
Government, higher education, insurance and tourism are principal businesses in Montpelier.

Industry
Since the city's establishment as capital in 1805, the primary business in Montpelier has been state government. By the mid-19th century, Montpelier was also the home of life and fire insurance companies including National Life Group.

The majority of businesses in the downtown area, mostly retail, are locally owned. These include antiques, books, handmade toys, crafts, kitchenware, jewelry, and Vermont-themed gifts.

The Savoy Theater, which was opened in 1905, is an eclectic art house with a main theater that holds 120 and a smaller room that holds 35. It specializes in documentaries, independent features, and international films. The annual Green Mountain Film Festival is held in Montpelier, and is hosted by the Savoy.

Tourism
The Vermont History Museum, operated in The Pavilion by the Vermont Historical Society, is an attraction.

Arts and culture

 * Kellogg-Hubbard Library—with a copy of the Parthenon Frieze
 * Lost Nation Theater
 * Montpelier Theatre Guild
 * Vermont History Museum—in The Pavilion
 * Vermont State House
 * T. W. Wood Gallery & Arts Center
 * Capital City Concerts
 * An annual local vernacular culture phenomenon, the Valentine Phantom, a tradition of covering downtown storefronts and public buildings with red hearts each February 14, began in Montpelier in the 1990s.

Sports
The Vermont Mountaineers of the New England Collegiate Baseball League play at the Montpelier Recreation Field.

The Grand Depart of the Vermont Super 8 bikepacking route takes place on the steps of the capitol each fall.

Parks and recreation
The city has three city nature centers. Hubbard Park rises behind the state capitol building and extends along the ridge line towards the north past the pool to the stump dump. Accessible from Cummings Street off State Route 12, the North Branch River Park is the second-largest park in the city. The Mill Pond Park is located along State Route 12 approximately 0.25 mi from the cemetery and features boat access to the North Branch river, as well as benches and short-term parking. The North Branch Nature Center is located at the northern end of town and includes 17 acre of protected land as well as a community nature center. A bridge from the North Branch Nature Center connects the land to the North Branch River Park on the opposite side of the North Branch River. In recent years, North Branch River Park has significantly expanding its mountain biking trails, making it a hub for area mountain bikers of all ages. The first mountain bike trail in North Branch River Park was constructed and opened in 2005 as part of a partnership between the Montpelier Area Mountain Bike Association and the Montpelier Parks Department. Since that time, 4 additional miles of trails have been added. A pump track is also currently under construction.

Government
Montpelier's government maintains a city council, city manager, and mayor. The city council consists of a mayor and six members. Council members are elected on a nonpartisan ballot from three districts, with councilors serving staggered two-year terms. The mayor is elected to a two-year term in each even-numbered year on a nonpartisan ballot in a citywide election. The council appoints the city manager who is the chief administrative officer of the city.

The city provides municipal services for its residents and businesses. These include local law enforcement, firefighting, planning and zoning regulation, and provision for potable drinking water and wastewater.

Politically, Montpelier is a Democratic stronghold. In the 2020 United States presidential election, Joe Biden received 88 percent of the vote to Donald Trump's 9 percent.

Education

 * Public schools include:
 * Montpelier High School
 * Main Street Middle School
 * Union Elementary School
 * River Rock School is a private school serving kindergarten through 8th grade students.
 * A campus of the Community College of Vermont
 * Union Institute & University Vermont campus, offers a Master of Education program through a low-residency (online) program
 * Vermont College of Fine Arts is a low-residency graduate school offering Masters of Fine Arts degrees in visual arts, writing, and writing for children and young adults

Media
The Barre Montpelier Times Argus is a daily newspaper that serves Montpelier and Washington County. The Montpelier Bridge is a semi-monthly paper covering the Montpelier area. The broadcast television station WNNE (The CW) is licensed to the city. ORCA Media is a non-profit community media center that serves Montpelier and surrounding communities, including Randolph and Waterbury.

Transportation
Montpelier has become one of Vermont's most readily accessible cities and towns, as Vermont's founders deliberately placed the capital near the geographic center of the state.

Roads
The city is located along Interstate 89. East–west U.S. Route 2 and north–south Vermont Route 12 are two other principal routes that intersect in Montpelier. Both I-89 and U.S. 2 provide a direct link to Burlington and the populous Lake Champlain Valley in the northwestern corner of the state. U.S. Route 302 has its western terminus in Montpelier, connecting it with the nearby city of Barre and points east.

Rail
Amtrak, the national passenger rail system, provides daily service from its station at Montpelier Junction in the neighboring town of Berlin, on the route known as the "Vermonter", operating between St. Albans, Vermont and Washington, D.C.

Bus
Greyhound and Megabus operate buses that serve Montpelier. The Green Mountain Transit Authority (GMTA) operates a local bus network throughout the micropolitan area, with stops in Montpelier and Barre, including nearby Waterbury, the Vermont State House, the Ben & Jerry's factory, and the local Berlin Mall. GMTA and its sister bus company in Burlington, the Chittenden County Transportation Authority (CCTA), operate a series of LINK commuter buses with stops in Montpelier, Burlington, Richmond, and Waterbury.

Air
Air travelers in private planes can use the Edward F. Knapp State Airport in Berlin to access Montpelier. The Burlington International Airport in Chittenden County is the closest commercial air service, located 35 mi northwest of Montpelier.

Other
Two shared-use paths for walking and bicycling connect to Montpelier: the Cross Vermont Trail and the Central Vermont Regional Path. Montpelier's downtown is relatively compact and pedestrian-friendly, with sidewalks and crosswalks throughout the downtown area.

Sister cities

 * 🇲🇽 Zapopan, Mexico

title: Montpelier, Vermont


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Intensity (physics)

In physics, the intensity or flux density of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m2), or kg⋅s−3 in base units. Intensity is used most frequently with waves such as acoustic waves (sound) or electromagnetic waves such as light or radio waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

The word "intensity" as used here is not synonymous with "strength", "amplitude", "magnitude", or "level", as it sometimes is in colloquial speech.

Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e., surface power density). The intensity of a wave is proportional to the square of its amplitude. For example, the intensity of an electromagnetic wave is proportional to the square of the wave's electric field amplitude.

Mathematical description
If a point source is radiating energy in all directions (producing a spherical wave), and no energy is absorbed or scattered by the medium, then the intensity decreases in proportion to the distance from the object squared. This is an example of the inverse-square law.

Applying the law of conservation of energy, if the net power emanating is constant, $$P = \int \mathbf I\, \cdot d\mathbf A,$$ where
 * $P$ is the net power radiated;
 * $I$ is the intensity vector as a function of position;
 * the magnitude $|I|$ is the intensity as a function of position;
 * $dA$ is a differential element of a closed surface that contains the source.

If one integrates a uniform intensity, $|I| = const.$, over a surface that is perpendicular to the intensity vector, for instance over a sphere centered around the point source, the equation becomes $$P = |I| \cdot A_\mathrm{surf} = |I| \cdot 4\pi r^2,$$ where
 * $|I|$ is the intensity at the surface of the sphere;
 * $r$ is the radius of the sphere;
 * $$A_\mathrm{surf} = 4\pi r^2 $$ is the expression for the surface area of a sphere.

Solving for $|I|$ gives $$|I| = \frac{P}{A_\mathrm{surf}} = \frac{P}{4\pi r^2}. $$

If the medium is damped, then the intensity drops off more quickly than the above equation suggests.

Anything that can transmit energy can have an intensity associated with it. For a monochromatic propagating electromagnetic wave, such as a plane wave or a Gaussian beam, if $E$ is the complex amplitude of the electric field, then the time-averaged energy density of the wave, travelling in a non-magnetic material, is given by: $$\left\langle U \right \rangle = \frac{n^2 \varepsilon_0}{2} |E|^2,$$ and the local intensity is obtained by multiplying this expression by the wave velocity, $\tfrac{\mathrm c}{n} \! :$ $$I = \frac{\mathrm{c} n \varepsilon_0}{2} |E|^2,$$ where
 * $n$ is the refractive index;
 * $c$ is the speed of light in vacuum;
 * $ε0$ is the vacuum permittivity.

For non-monochromatic waves, the intensity contributions of different spectral components can simply be added. The treatment above does not hold for arbitrary electromagnetic fields. For example, an evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity should then be defined as the magnitude of the Poynting vector.

Alternative definitions
In photometry and radiometry intensity has a different meaning: it is the luminous or radiant power per unit solid angle. This can cause confusion in optics, where intensity can mean any of radiant intensity, luminous intensity or irradiance, depending on the background of the person using the term. Radiance is also sometimes called intensity, especially by astronomers and astrophysicists, and in heat transfer.

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

title: Intensity (physics)


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Talk:Möbius strip

Twice cut strip
Wouldn't that create two Möbius strips instead? Explanation caption for the image lists that it creates a Möbius strip with two tracks, and one non Möbius strip. In reality, this should create a two tracked Möbius strip and a separate Möbius strip. Cmdrscotty (talk) 14:21, 1 September 2020 (UTC)

 * The pink/blue strip has two sides and is not a Möbiusstrip. --Ag2gaeh (talk) 16:41, 1 September 2020 (UTC)

Mobius M. Mobius
Should Mobius M. Mobius be included in the popular culture section? (Sources: 1, 2, 3, 4, 5) Sahaib (talk) 17:35, 3 May 2022 (UTC)
 * I'm leaning towards no. If the connection is only to the misspelled name "Mobius", no. If we have in-depth published reliable sources specifically connecting this character's plot line to the Möbius strip itself, and not just fan speculation which some of your sources look like, and this plot line is the main point of a movie about this character, then maybe it could be listed among other fictional works with Möbius-strip plotlines. Or if your links' theory that this is another example of the bullshit "Möbius strips are things we don't understand therefore they could cause time loops" trope turns out to be validated, then maybe it could be mentioned, briefly, in the sentence with three examples of this trope. But the current Wikipedia article Mobius M. Mobius contains no content that would make a significant enough connection to this article to link to. The popular culture section is not intended as a catch-all listing of anything vaguely related (go to tvtropes for that sort of thing), but as an organized survey of different wide areas of popular culture to which the idea of a Möbius strip has had a significant influence (fine art, design, literature, etc). A name of a character in a movie franchise is not a significant influence to a wide area of popular culture. We already have plenty of examples of movies where the strip plays a much more significant role in the overall structure of the movie and not just in the names or behavior of one its characters. —David Eppstein (talk) 17:43, 3 May 2022 (UTC)

Area of swept surface
It might be interesting to say what the area of the surface swept by a rotating line segment is. I don't know the answer, but I found a reference saying what the area isn't: it isn't what you would get by trying to apply Pappus's centroid theorem. The rotation makes it sweep out a bigger area than a non-rotating segment would. See the last line of: —David Eppstein (talk) 01:21, 4 May 2022 (UTC)
 * Interesting. Numerically (from their formula) it's ≈6.62715 for a fairly natural choice of dimensions, namely a strip width of 1 unit rotating while following a circle of radius 1, indeed slightly greater than Pappus's theorem's estimate of 2π. I searched for that constant (and approximations) and couldn't find anything, which is a crude way of going about things, but I'd find it highly unlikely that someone would evaluate the area analytically and not put a numerical value to it. So I suspect it's not been properly investigated, and I doubt there's a closed form. For posterity, the integral is $$\int_0^{2\pi}\int_0^1 \sqrt{(t/2)^2+(1+t \cos(s/2))^2}\, dt\,ds$$. Ovinus (talk) 03:26, 9 May 2022 (UTC)

Relation to Umbilic torus
The link to "Umbilic torus" in the "See also" section is accompanied with a claim that the Umbilic torus can be obtained from a Möbius strip by gluing the latter along its single edge. IMHO, that statement is false, at least in regular 3D space, where the result would have to be a self-intersecting surface, assuming that the gluing is performed in the 3D space, rather than through some topological magic (sorry, I am not an expert in this field and lack the right terminology; my judgements are restricted to common sense intuition). Note that the article about the umbilic torus doesn't establish any relation of it with the Möbius strip (the link in the "See also" section doesn't count as such). I think that the misleading sentence has to be either removed or reformulated in a less confusing way and a more detailed explanation can be provided in the article about umbilic torus. Leon.Manukyan (talk) 08:41, 23 September 2023 (UTC)

 * You are confusing the standard embedding of the Möbius strip (with one half-twist) with the Möbius strip itself (a topological surface independent of how it is embedded into space). If you perform any odd number of half-twists to a strip before gluing it back to itself, you get a Möbius strip, with a different embedding for each different number of half-twists (see the second paragraph of the lead). The umbilic torus has a boundary that, if you cut along the ridges, is a Möbius strip. It is a Möbius strip embedded with three half-twists rather than one, but that is still a Möbius strip. —David Eppstein (talk) 10:15, 23 September 2023 (UTC)
 * You provided the same justification in your reversal of my earlier edit. My feeling was that something was wrong with it. A Möbius strip is single-sided whereas the Umbilic torus apparently has inside and outside. Now thinking about it more carefully, I guess that I found the flaw in your reasoning. The fact that the boundary goes three times around the center of the torus doesn't mean that the strip makes three half-twists. Actually it only makes a full twist (two half-twists). Hence it is not topologically equivalent to a Möbius strip. Leon.Manukyan (talk) 10:31, 23 September 2023 (UTC)
 * Ok, this time I think you may be correct. If it really were a Möbius strip then (regardless of number of twists) traveling around the boundary (without crossing any ridges) until you return to your start should take you to the inside, but it doesn't. —David Eppstein (talk) 11:53, 23 September 2023 (UTC)

title: Talk:Möbius strip


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Twin prime

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair $(17, 19)$ or $(41, 43)$. In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.

Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.

Properties
Usually the pair $(2, 3)$ is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.

The first several twin prime pairs are

Five is the only prime that belongs to two pairs, as every twin prime pair greater than $(3, 5), (5, 7), (11, 13),$ is of the form $$(6n-1, 6n+1)$$ for some natural number $n$; that is, the number between the two primes is a multiple of 6. As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.

Brun's theorem
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than $N$ does not exceed
 * $$\frac{CN}{(\log N)^2}$$

for some absolute constant $C$ > 0. In fact, it is bounded above by $$\frac{8 C_2 N}{(\log N)^2} \left[ 1 + \operatorname{\mathcal O}\left(\frac{\log \log N}{\log N} \right) \right],$$ where $$C_2$$ is the twin prime constant (slightly less than 2/3), given below.

Twin prime conjecture
The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states that there are infinitely many primes $p$ such that $(17, 19), (29, 31), (41, 43),$ is also prime. In 1849, de Polignac made the more general conjecture that for every natural number $k$, there are infinitely many primes $p$ such that $(59, 61), (71, 73), (101, 103),$ is also prime. The case $er$ = 1 of de Polignac's conjecture is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the prime number theorem.

On 17 April 2013, Yitang Zhang announced a proof that for some integer $k$ that is less than 70 million, there are infinitely many pairs of primes that differ by $N$. Zhang's paper was accepted in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound.

As of 14 April 2014, one year after Zhang's announcement, the bound has been reduced to 246. These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao. This second approach also gave bounds for the smallest $(107, 109), (137, 139), ...$ needed to guarantee that infinitely many intervals of width $(3, 5)$ contain at least $N$ primes. Moreover (see also the next section) assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath Project wiki states that the bound is 12 and 6, respectively.

A strengthening of Goldbach’s conjecture, if proved, would also prove there is an infinite number of twin primes, as would the existence of Siegel zeroes.

Other theorems weaker than the twin prime conjecture
In 1940, Paul Erdős showed that there is a constant $6n+1$ and infinitely many primes $m$ such that $6n−1$ where $p$ denotes the next prime after $p′$. What this means is that we can find infinitely many intervals that contain two primes $p + 2$ as long as we let these intervals grow slowly in size as we move to bigger and bigger primes. Here, "grow slowly" means that the length of these intervals can grow logarithmically. This result was successively improved; in 1986 Helmut Maier showed that a constant $p + 2k$ can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to $f (m)$. In 2005, Goldston, Pintz, and Yıldırım established that $p$ can be chosen to be arbitrarily small, i.e.
 * $$\liminf_{n\to\infty} \left( \frac{ p_{n+1} - p_n }{\log p_n} \right) = 0 ~.$$

On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example, $f (m)$.

By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many $c$ such that at least two of $n$, $c < 1$, $p′ − p < c ln p$, $(p, p′)$, $c < 0.25$, $c = 0.085786...$, or $c ln ln p$ are prime. Under a stronger hypothesis they showed that for infinitely many $n$, at least two of $n$, $n + 2$, $n + 6$, and $n + 8$ are prime.

The result of Yitang Zhang,

 * $$ \liminf_{n\to\infty} (p_{n+1} - p_n) < N ~ \mathrm{ with } ~ N=7 \times 10^7,$$

is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard have reduced the bound: the limit inferior is at most 246.

First Hardy–Littlewood conjecture
The first Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let $n$ denote the number of primes $n + 12$ such that $n + 18$ is also prime. Define the twin prime constant $n + 20$ as $$ C_2 = \prod_{\textstyle{p \; \mathrm{prime,}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2} \right) \approx 0.66016 18158 46869 57392 78121 10014 \ldots. $$ (Here the product extends over all prime numbers $n + 2$.) Then a special case of the first Hardy-Littlewood conjecture is that $$ \pi_2(x) \sim 2 C_2 \frac{x}{(\ln x)^2} \sim 2 C_2 \int_2^x {\mathrm{d} t \over (\ln t)^2} $$ in the sense that the quotient of the two expressions tends to 1 as $\pi_2(x)$ approaches infinity. (The second ~ is not part of the conjecture and is proven by integration by parts.)

The conjecture can be justified (but not proven) by assuming that $x$ describes the density function of the prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for $\tfrac{1}{\ln t}$ above.

The fully general first Hardy–Littlewood conjecture on prime $\pi_2(x)$-tuples (not given here) implies that the second Hardy–Littlewood conjecture is false.

This conjecture has been extended by Dickson's conjecture.

Polignac's conjecture
Polignac's conjecture from 1849 states that for every positive even integer $k$, there are infinitely many consecutive prime pairs $k$ and $p$ such that $n + 4$ (i.e. there are infinitely many prime gaps of size $p′$). The case $n + 6$ is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of $k$, but Zhang's result proves that it is true for at least one (currently unknown) value of $k$. Indeed, if such a $k$ did not exist, then for any positive even natural number $k$ there are at most finitely many $N$ such that $$p_{n+1} - p_n = m$$ for all $p ≤ x$ and so for $n$ large enough we have $$p_{n+1} - p_n > N,$$ which would contradict Zhang's result.

Large twin primes
Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-largest twin primes. , the current largest twin prime pair known is 2996863034895 × 2$n$ ± 1, with 388,342 decimal digits. It was discovered in September 2016.

There are 808,675,888,577,436 twin prime pairs below.

An empirical analysis of all prime pairs up to 4.35 × shows that if the number of such pairs less than $1290000$ is $p + 2$ then $C2$ is about 1.7 for small $1290000$ and decreases towards about 1.3 as $n$ tends to infinity. The limiting value of $p ≥ 3$ is conjectured to equal twice the twin prime constant (not to be confused with Brun's constant), according to the Hardy–Littlewood conjecture.

Other elementary properties
Every third odd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a Chen prime.

It has been proven that the pair (m, m + 2) is a twin prime if and only if

 * $$4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.$$

If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.

For a twin prime pair of the form (6n − 1, 6n + 1) for some natural number n > 1, n must end in the digit 0, 2, 3, 5, 7, or 8.

Isolated prime
An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite.

The first few isolated primes are

 * 2, 23, 37, 47, 53, 67, 79, 83, 89, 97, ....

It follows from Brun's theorem that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold n and the number of all primes less than n tends to 1 as n tends to infinity.

title: Twin prime


text:
Talk:Brun's constant

Untitled
Axel: Brun's constant is denoted as B but Nicely (I don't know why) uses B2 so I don't see the meaning of change. Please see other references too. XJamRastafireRootsRockReggaeSecurityInvestigator [2002.02.27] 3 Wednesday somewhere outa space.

Well, you used both B and B2 in your article, and I decided that one name for the number is enough. If you prefer B, please change it here and also in mathematical constants. AxelBoldt

 * No need. Let it stay B2 and let future shows up the decision. Perhaps it would come out that we should name this constant Brun - Hauschaeckell's B3z constant. Who knows --XJam [2002.02.27] 3 Wednesday (2nd ed.)

Axel, is this better?

1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers p and q which differ by two) B2(p,q):

 * B2(p,q) = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + (1/41 + 1/43) + (1/59 + 1/61) + ...

converges to a finite constant now called Brun's constant for twin primes and thus usually denoted by B2 and defined as:

 * B2 = limp,q→∞ B2(p,q).

I think we should 'somehow' distinguish between the sum B2(p,q) and the Brun's constant B2.

XJam [2002.04.02] 2 Tuesday (0)

No, your notation is unclear: you write B2(p,q) for a number that doesn't depend on p and q! Your definition of B2(p,q) above is exactly Brun's constant; the limit is already built in because of the ... in the formula. It is an infinite series.

I think the definition in the article is clear right now. AxelBoldt, Tuesday, April 2, 2002

Ralf: The link to 'Pascal Sebah' leads to a 19th century photographer.

Bn generalizations
This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n. This gives the estimate for Bn of 2, about 5% higher than the true value. Firstly, it's not entirely clear at first glance whether the second sentence refers to the "cousin" type of Bn that was just defined, or if the two sentences (which were added at different times) are adjacent by coincidence and the second one is referring to the "quadruplet" type of Bn. (The answer seems to be that it's the "cousin" type -- neighboring primes with a difference of exactly n.) Secondly, I just looked at the abstract of Wolf's paper, and the formula is not 4/n, but rather something more complicated with a 4 in the numerator and an n in the denominator. Thirdly, Wolf's result only applies to n >= 6, so even if the 4/n formula is somehow correct, applying it to B2 strikes me as misleading or pointless. Fourthly, I'm a little bothered by calling 1.902 the "true value", when it's stated earlier that no upper bound for the value has been proved. I'm inclined to delete everything after the first sentence, but I'll leave it alone for now and see if anyone else agrees or disagrees. 76.202.61.191 (talk) 06:03, 19 May 2008 (UTC)

 * Can anyone please explain for me, briefly, why an irrational Brun's constant implies infinitude of twin primes? Thanks.209.167.89.139 (talk) 13:48, 10 August 2009 (UTC)

 * If there were a finite number of twin primes, then Brun's constant would be a finite sum of rational numbers, so it would be rational itself. So, if we find out that Brun's constant is irrational, that would mean that there isn't a finite number of twin primes. --Sfztang (talk) 05:17, 16 October 2009 (UTC)

Bound on B2
"While 1.9 < B2 is shown, no real number N is known such that B2 < N." What about N=2? Am I missing something? – Sigbhu —(talk • contribs) 15:10, 15 February 2010 (UTC)

 * In I have written about the estimate 1.902160583104:
 * "It is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. While 1.83 < B2 is shown, no real number N is known such that B2 < N.[Citation needed]"
 * It is my understanding it has only been proved that B2 is finite and at least 1.830484424658... which is the sum for twin primes below 1016. The rest is only a heuristic guess based on the expected frequency of twin primes above 1016. The twin primes below 1016 follow the expectation well and it appears likely that those above will also do it, but large twin primes could theoretically be common enough to make B2 much larger than 2, or rare enough to make it below 1.9. I have therefore removed the claim that the best estimate is 1.902160583104. Maybe it is true that it is the best estimate but it has not been proved to be the best (assuming "best" means closest to the real value). PrimeHunter (talk) 02:14, 16 February 2010 (UTC)

Crandall & Pomerance
Someone should get and read their book. There's no proof in it.Eganfan (talk) 12:41, 3 July 2010 (UTC)
 * So tag it as verification failed, then. Real bounds would be notable.  — Arthur Rubin  (talk) 18:14, 3 July 2010 (UTC)
 * No real bounds are known. Eganfan (talk) 21:22, 3 July 2010 (UTC)

 * I have reverted your removal with edit summary "Crandall & Pomerance is a reliable source and verifiably says B_2 < 2.347, accepted by others, Wikipedia does not require proofs in reliable sources". See also for example page 1 of http://www.math.uiuc.edu/~pppollac/polyapps.pdf which says: "The upper bound is due to Crandall and Pomerance ([5, pp. 16-17], see also [19, Chapter 3]), who bound the sum of the twin prime pairs past 1016 using an explicit upper estimate of Riesel and Vaughan [29] for the number of twin prime pairs." PrimeHunter (talk) 23:05, 3 July 2010 (UTC)
 * I don't see your point. All I'm saying is there is no proof of B_2 < 2.347. Eganfan (talk) 08:37, 4 July 2010 (UTC)
 * I have reverted you again. We don't need a proof in the cited source when the source is reliable. It is cited as a source for the claim "B_2 < 2.347", not for a mathematical proof of that claim. The citation satisfies Verifiability. Wikipedia uses the same citation rules for all the millions of articles. Few of them are about subjects where a mathematical proof of correctness is possible. We report what reliable sources say. If you can find another reliable source contradicting the respected Crandall & Pomerance then we can discuss it, but otherwise you should stop edit warring. PrimeHunter (talk) 21:21, 4 July 2010 (UTC)
 * What are you talking about? No one ever proved "B_2 < 2.347". Are you religious person or what?Eganfan (talk) 21:49, 4 July 2010 (UTC)

 * Clearly the reference is not ideal. It would be nice if it gave a proof or had a reference to the original source. Nevertheless it is a serious scientific source and I see no reason not to trust them. Especially since experts in this comment thread seemed convinced that one should be able to get some upper bound using Brun's method. I am surprised by Eganfan's confident insistence to the contrary. I tried asking on Mathoverflow but people there were not aware of any better references. MathHisSci (talk) 09:08, 10 October 2010 (UTC)

title: Talk:Brun's constant


text:
Franz Mertens

Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish mathematician. He was born in Schroda in the Grand Duchy of Posen, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Austria.

The Mertens function M(x) is the sum function for the Möbius function, in the theory of arithmetic functions. The Mertens conjecture concerning its growth, conjecturing it bounded by x1/2, which would have implied the Riemann hypothesis, is now known to be false (Odlyzko and te Riele, 1985). The Meissel–Mertens constant is analogous to the Euler–Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once. Mertens's theorems are three 1874 results related to the density of prime numbers.

Erwin Schrödinger was taught calculus and algebra by Mertens.

His memory is honoured by the Franciszek Mertens Scholarship granted (from 2017) to those outstanding pupils of foreign secondary schools who wish to study at the Faculty of Mathematics and Computer Science of the Jagiellonian University in Kraków and were finalists of the national-level mathematics, or computer science olympiads, or they have participated in one of the following international olympiads: in mathematics (IMO), computer science (IOI), artificial intelligence (IOAI), astronomy (IAO), astronomy and astrophysics (IOAA), physics (IPhO), linguistics (IOL), European Girls' Mathematical Olympiad (EGMO), European Girls’ Olympiad in Informatics (EGOI), Romanian Masters of Mathematics (RMM), Romanian Masters of Informatics (RMI) or International Zhautykov Olympiad (IZhO).

title: Franz Mertens


text:
Émile Picard

Charles Émile Picard (24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.

Life
He was born in Paris on 24 July 1856 and educated there at the Lycée Henri-IV. He then studied mathematics at the École Normale Supérieure.

Picard's mathematical papers, textbooks, and many popular writings exhibit an extraordinary range of interests, as well as an impressive mastery of the mathematics of his time. Picard's little theorem states that every nonconstant entire function takes every value in the complex plane, with perhaps one exception. Picard's great theorem states that an analytic function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity. He made important contributions in the theory of differential equations, including work on Picard–Vessiot theory, Painlevé transcendents and his introduction of a kind of symmetry group for a linear differential equation. He also introduced the Picard group in the theory of algebraic surfaces, which describes the classes of algebraic curves on the surface modulo linear equivalence. In connection with his work on function theory, he was one of the first mathematicians to use the emerging ideas of algebraic topology. In addition to his theoretical work, Picard made contributions to applied mathematics, including the theories of telegraphy and elasticity. His collected papers run to four volumes.

Louis Couturat studied integral calculus with Picard in 1891-1892, taking detailed notes of the lectures. These notes were preserved and now are available in three cahiers from Internet Archive.

Like his contemporary, Henri Poincaré, Picard was much concerned with the training of mathematics, physics, and engineering students. He wrote a classic textbook on analysis and one of the first textbooks on the theory of relativity. Picard's popular writings include biographies of many leading French mathematicians, including his father in law, Charles Hermite.

Picard was an International Honorary Member of the American Academy of Arts and Sciences, an International Member of the United States National Academy of Sciences, and an International Member of the American Philosophical Society.

Family
In 1881 he married Marie, the daughter of Charles Hermite.

Society
Charles Émile Picard was one of the founder members in 1930 of the Society of Friends of André-Marie Ampère which was created to develop the first science museum in France, the Ampère Museum close to Lyons.

Works

 * 1891–96:
 * 1905:
 * 1906 : (with Georges Simart) Theorie des Fonctions Algebrique de deux Variables Independente volume 2, via Internet Archive
 * 1922:
 * 1922:
 * 1931:
 * 1978–81:

title: Émile Picard


text:
Antonie van Leeuwenhoek

Antonie Philips van Leeuwenhoek (24 October 1632 – 26 August 1723) was a Dutch microbiologist and microscopist in the Golden Age of Dutch science and technology. A largely self-taught man in science, he is commonly known as "the Father of Microbiology", and one of the first microscopists and microbiologists. Van Leeuwenhoek is best known for his pioneering work in microscopy and for his contributions toward the establishment of microbiology as a scientific discipline.

Raised in Delft, Dutch Republic, Van Leeuwenhoek worked as a draper in his youth and founded his own shop in 1654. He became well-recognized in municipal politics and developed an interest in lensmaking. In the 1670s, he started to explore microbial life with his microscope.

Using single-lensed microscopes of his own design and make, Van Leeuwenhoek was the first to observe and to experiment with microbes, which he originally referred to as dierkens, diertgens or diertjes. He was the first to relatively determine their size. Most of the "animalcules" are now referred to as unicellular organisms, although he observed multicellular organisms in pond water. He was also the first to document microscopic observations of muscle fibers, bacteria, spermatozoa, red blood cells, crystals in gouty tophi, and among the first to see blood flow in capillaries. Although Van Leeuwenhoek did not write any books, he described his discoveries in chaotic letters to the Royal Society, which published many of his letters in their Philosophical Transactions.

Early life and career

Antonie van Leeuwenhoek was born in Delft, Dutch Republic, on 24 October 1632. On 4 November, he was baptized as Thonis. His father, Philips Antonisz van Leeuwenhoek, was a basket maker who died when Antonie was only five years old. His mother, Margaretha (Bel van den Berch), came from a well-to-do brewer's family. She remarried Jacob Jansz Molijn, a painter and the family moved to Warmond around 1640. Antonie had four older sisters: Margriet, Geertruyt, Neeltje, and Catharina. When he was around ten years old his step-father died. He was sent to live in Benthuizen with his uncle, an attorney. At the age of 16 he became a bookkeeper's apprentice (casher) at a linen-draper's shop at Warmoesstraat in Amsterdam, which was owned by William Davidson. Van Leeuwenhoek left there after six years.

In July 1654 Van Leeuwenhoek married Barbara de Mey in Delft, with whom he fathered one surviving daughter, Maria (four other children died in infancy). He would live and study for the rest of his life at Hypolytusbuurt in a house he bought in 1655. He opened a draper's shop, selling linen, yarn and ribbon to seamstresses and tailors. His status in Delft grew throughout the years. In 1660 he received a lucrative job as chamberlain for the sheriffs in the city hall, a position which he would hold for almost 40 years. His duties included maintaining the premises, heating, cleaning, opening for meetings, performing duties for those assembled, and maintaining silence on all matters discussed there.

In 1669 he was appointed as a land surveyor by the court of Holland; at some time he combined it with another municipal job, being the official "wine-gauger" of Delft and in charge of the city wine imports and taxation. His wife had died in 1666, and in 1671, Van Leeuwenhoek remarried to Cornelia Swalmius with whom he had no children.

Van Leeuwenhoek was a contemporary of another famous Delft citizen, the painter Johannes Vermeer, who was baptized just four days earlier. It has been suggested that he is the man portrayed in two Vermeer paintings of the late 1660s, The Astronomer and The Geographer, but others argue that there appears to be little physical similarity. Because they were both relatively important men in a city with only 24,000 inhabitants, living both close to the main market, it is likely they knew each other. Van Leeuwenhoek acted as the executor of Vermeer's will when the painter died in 1675.

Van Leeuwenhoek's religion was "Dutch Reformed" and Calvinist. Like Jan Swammerdam he often referred with reverence to the wonders God designed in making creatures great and small, and believed that his discoveries were merely further proof of the wonder of creation.

Microscopic study

While running his draper shop, Van Leeuwenhoek wanted to see the quality of the thread better than what was possible using the magnifying lenses of the time. He developed an interest in lensmaking, although few records exist of his early activity. By placing the middle of a small rod of soda lime glass in a hot flame, one can pull the hot section apart to create two long whiskers of glass. Then, by reinserting the end of one whisker into the flame, a very small, high-quality glass lens is created. Significantly, a May 2021 neutron tomography study of a high-magnification Leeuwenhoek microscope captured images of the short glass stem characteristic of this lens creation method. For lower magnifications he also made ground lenses. To help keep his methods confidential he apparently intentionally encouraged others to think grinding was his primary or only lens construction method.

Recognition by the Royal Society
After developing his method for creating powerful lenses and applying them to the study of the microscopic world, Van Leeuwenhoek introduced his work to his friend, the prominent Dutch physician Reinier de Graaf. When the Royal Society in London published the groundbreaking work of an Italian lensmaker in their journal Philosophical Transactions of the Royal Society, de Graaf wrote to the editor of the journal, Henry Oldenburg, with a ringing endorsement of Van Leeuwenhoek's microscopes which, he claimed, "far surpass those which we have hitherto seen". In response, in 1673 the society published a letter from Van Leeuwenhoek that included his microscopic observations on mold, bees, and lice. Then, in 1674, Van Leeuwenhoek made his most significant discovery. Starting from the assumption that life and motility are similar, he determined that the moving objects observed under his microscope were little animals. He later recorded his observations in his diary.

Van Leeuwenhoek's work fully captured the attention of the Royal Society, and he began corresponding regularly with the society regarding his observations. At first he had been reluctant to publicize his findings, regarding himself as a businessman with little scientific, artistic, or writing background, but de Graaf urged him to be more confident in his work. By the time Van Leeuwenhoek died in 1723, he had written some 190 letters to the Royal Society, detailing his findings in a wide variety of fields, centered on his work in microscopy. He only wrote letters in his own colloquial Dutch; he never published a proper scientific paper in Latin. He strongly preferred to work alone, distrusting the sincerity of those who offered their assistance. The letters were translated into Latin or English by Henry Oldenburg, who had learned Dutch for this very purpose. He was also the first to use the word animalcules to translate the Dutch words that Leeuwenhoek used to describe microorganisms. Despite the initial success of Van Leeuwenhoek's relationship with the Royal Society, soon relations became severely strained. His credibility was questioned when he sent the Royal Society a copy of his first observations of microscopic single-celled organisms dated 9 October 1676. Previously, the existence of single-celled organisms was entirely unknown. Thus, even with his established reputation with the Royal Society as a reliable observer, his observations of microscopic life were initially met with some skepticism.

Eventually, in the face of Van Leeuwenhoek's insistence, the Royal Society arranged for Alexander Petrie, minister to the English Reformed Church in Delft; Benedict Haan, at that time Lutheran minister at Delft; and Henrik Cordes, then Lutheran minister at the Hague, accompanied by Sir Robert Gordon and four others, to determine whether it was in fact Van Leeuwenhoek's ability to observe and reason clearly, or perhaps, the Royal Society's theories of life that might require reform. Finally in 1677, Van Leeuwenhoek's observations were fully acknowledged by the Royal Society.

Antonie van Leeuwenhoek was elected to the Royal Society in February 1680 on the nomination of William Croone, a then-prominent physician. Van Leeuwenhoek was "taken aback" by the nomination, which he considered a high honour, although he did not attend the induction ceremony in London, nor did he ever attend a Royal Society meeting. He had his portrait painted by Jan Verkolje with the certificate signed by James II of England on the table beside him.

Scientific fame
By the end of the seventeenth century, Van Leeuwenhoek had a virtual monopoly on microscopic study and discovery. His contemporary Robert Hooke, an early microscope pioneer, bemoaned that the field had come to rest entirely on one man's shoulders. In 1673, his first letter was published in the journal of the Royal Society of London. He was visited over the years by many notable individuals who gazed at the tiny creatures. One of the first was Jan Swammerdam. Around 1675 it was Johan Huydecoper, who was very interested in collecting and growing plants for his estate Goudestein, becoming in 1682 manager of the Hortus Botanicus Amsterdam. Christiaan Huygens, Leibniz (1676), John Locke (1678, 1685), James II of England (1679), William III of Orange, Mary II of England and Thomas Molyneux (in 1685) visited. In October 1697, Van Leeuwenhoek visited the Tsar Peter the Great on his boat, moored in the Schie or the Arsenaal. On this occasion he presented the Tsar with an "eel-viewer", so Peter could study blood circulation whenever he wanted. In 1706 it was Govert Bidloo; in 1714 Richard Bradley (botanist), in 1716 Herman Boerhaave and Frederik Ruysch. To the disappointment of his guests, Van Leeuwenhoek refused to reveal the cutting-edge microscopes he relied on for his discoveries, instead showing visitors a collection of average-quality lenses.

Techniques

Antonie van Leeuwenhoek made more than 500 optical lenses. He also created at least 25 single-lens microscopes, of differing types, of which only nine have survived. These microscopes were made of silver or copper frames, holding hand-made lenses. Those that have survived are capable of magnification up to 275 times. It is suspected that Van Leeuwenhoek possessed some microscopes that could magnify up to 500 times. Although he has been widely regarded as a dilettante or amateur, his scientific research was of remarkably high quality.

The single-lens microscopes of Van Leeuwenhoek were relatively small devices, the largest being about 5 cm long. They are used by placing the lens very close in front of the eye. The other side of the microscope had a pin, where the sample was attached in order to stay close to the lens. There were also three screws to move the pin and the sample along three axes: one axis to change the focus, and the two other axes to navigate through the sample.

Van Leeuwenhoek maintained throughout his life that there are aspects of microscope construction "which I only keep for myself", in particular his most critical secret of how he made the lenses. For many years no one was able to reconstruct Van Leeuwenhoek's design techniques, but in 1957, C. L. Stong used thin glass thread fusing instead of polishing, and successfully created some working samples of a Van Leeuwenhoek design microscope. Such a method was also discovered independently by A. Mosolov and A. Belkin at the Russian Novosibirsk State Medical Institute. In May 2021 researchers in the Netherlands published a non-destructive neutron tomography study of a Leeuwenhoek microscope. One image in particular shows a Stong/Mosolov-type spherical lens with a single short glass stem attached (Fig. 4). Such lenses are created by pulling an extremely thin glass filament, breaking the filament, and briefly fusing the filament end. The nuclear tomography article notes this lens creation method was first devised by Robert Hooke rather than Leeuwenhoek, which is ironic given Hooke's subsequent surprise at Leeuwenhoek's findings.

Van Leeuwenhoek used samples and measurements to estimate numbers of microorganisms in units of water. He also made good use of the huge advantage provided by his method. He studied a broad range of microscopic phenomena, and shared the resulting observations freely with groups such as the British Royal Society. Such work firmly established his place in history as one of the first and most important explorers of the microscopic world. Van Leeuwenhoek was one of the first people to observe cells, much like Robert Hooke. He also corresponded with Antonio Magliabechi.

Discoveries

Van Leeuwenhoek has been recognized as the first person to use a histological stain to color specimens observed under the microscope using saffron. He used this technique only once.
 * Leeuwenhoek was one of the first to conduct experiments on himself. It was from his finger that blood was drawn for examination, and he placed pieces of his skin under a microscope, examining its structure in various parts of the body, and counting the number of vessels that permeate it.
 * Both Marcello Malpighi and Jan Swammerdam saw these structures before Leeuwenhoek, but Leeuwenhoek was the first to recognize what they are: red blood cells.
 * infusoria (protists in modern zoological classification), in 1674
 * in 1675 he was studying a variety of minerals, especially salts, and parts of plants and animals.
 * the vacuole of the cell in 1676
 * spermatozoa, in 1677
 * the banded pattern of muscular fibers, in 1682
 * bacteria, (e.g., large Selenomonads from the human mouth), in 1683
 * It seems he used horseradish to find out what causes irritation on the tongue. He used the effect of vinegar.
 * Leeuwenhoek diligently began to search for his animalcules. He found them everywhere: in rotten water, in ditches, on his own teeth. "Although I am now fifty years old," he wrote to the Royal Society, "my teeth are well preserved, because I am in the habit of rubbing them with salt every morning." He described paradontitis.
 * In 1684 he published his research on the ovary.
 * In 1687, Van Leeuwenhoek reported his research on the coffee bean. He roasted the bean, cut it into slices and saw a spongy interior. The bean was pressed, and an oil appeared. He boiled the coffee with rain water twice and set it aside.
 * Leeuwenhoek corresponded regularly with Anthonie Heinsius, the Delft pensionary in the States of Holland and in 1687 member of the board of the Delft chamber of the VOC.
 * In 1696 Nicolaas Witsen send him a map of Tartary and ore found near the Amur in Siberia.

 * In 1702 he requested a book on Peruvian silver mines in Potosí.

Like Robert Boyle and Nicolaas Hartsoeker, Van Leeuwenhoek was interested in dried cochineal, trying to find out if the dye came from a berry or an insect.

He studied rainwater, the seeds of oranges, worms in sheep's liver, the eye of a whale, the blood of fishes, mites, coccinellidae, the skin of elephants, Celandine, and Cinchona.

Legacy and recognition

By the end of his life, Van Leeuwenhoek had written approximately 560 letters to the Royal Society and other scientific institutions concerning his observations and discoveries. Even during the last weeks of his life, Van Leeuwenhoek continued to send letters full of observations to London. The last few contained a precise description of his own illness. He suffered from a rare disease, an uncontrolled movement of the midriff, which now is named van Leeuwenhoek's disease. He died at the age of 90, on 26 August 1723, and was buried four days later in the Oude Kerk in Delft.

In 1981, the British microscopist Brian J. Ford found that Van Leeuwenhoek's original specimens had survived in the collections of the Royal Society of London. They were found to be of high quality, and all were well preserved. Ford carried out observations with a range of single-lens microscopes, adding to our knowledge of Van Leeuwenhoek's work. In Ford's opinion, Leeuwenhoek remained imperfectly understood, the popular view that his work was crude and undisciplined at odds with the evidence of conscientious and painstaking observation. He constructed rational and repeatable experimental procedures and was willing to oppose received opinion, such as spontaneous generation, and he changed his mind in the light of evidence.

On his importance in the history of microbiology and science in general, the British biochemist Nick Lane wrote that he was "the first even to think of looking—certainly, the first with the power to see." His experiments were ingenious, and he was "a scientist of the highest calibre", attacked by people who envied him or "scorned his unschooled origins", not helped by his secrecy about his methods.

The Antoni van Leeuwenhoek Hospital in Amsterdam, named after Van Leeuwenhoek, is specialized in oncology. In 2004, a public poll in the Netherlands to determine the greatest Dutchman ("De Grootste Nederlander") named Van Leeuwenhoek the 4th-greatest Dutchman of all time.

On 24 October 2016, Google commemorated the 384th anniversary of Van Leeuwenhoek's birth with a Doodle that depicted his discovery of "little animals" or animalcules, now known as unicellular organisms.

The Leeuwenhoek Medal, Leeuwenhoek Lecture, Leeuwenhoek crater, Leeuwenhoeckia, Levenhookia (a genus in the family Stylidiaceae), Leeuwenhoekiella (an aerobic bacterial genus), and the scientific publication Antonie van Leeuwenhoek: International Journal of General and Molecular Microbiology are named after him.

title: Antonie van Leeuwenhoek


text:
Talk:Antonie van Leeuwenhoek

Leeuwenhoek
Please tell me pronounciation 106.216.161.49 (talk) 12:36, 23 February 2023 (UTC)

Wrong word—
In the following sentence: "Van Leeuwenhoek has been attributed as the first person to use a histological stain to color specimens observed under the microscope using saffron."

The word "attributed" is improperly used. "Acknowledged" would be more appropriate here. —Or perhaps the wording could be changed to: "The first use of a histological stain (he used saffron) to color specimens under the microscope has been attributed to van Leeuwenhoek." 2001:5B0:270F:3228:3147:99FB:E9E0:82FA (talk) 16:46, 6 November 2023 (UTC)

 * B..... 117.236.131.39 (talk) 14:07, 14 April 2024 (UTC)

Chaotic letters
it would be more respectful to say "long" letters Eugenio.orsi (talk) 21:59, 7 January 2024 (UTC)

Inconsistent use of capital letters for "van"
Should it be "van Leeuwenhoek" or "Van Leeuwenhoek"? I think it should consistently be spelled "van Leeuwenhoek", but the article is inconsistent. Bueller 007 (talk) 22:27, 28 March 2024 (UTC)

 * It should be "Van Leeuwenhoek" when his first name is not mentioned (which he himself spelled as Antoni, btw, without the E at the end), and "van Leeuwenhoek" when his first name is mentioned. So it's either "Antoni van Leeuwenhoek crossed the road", or "Van Leeuwenhoek crossed the road".
 * Either way, they keep omitting the "van" or "Van" from his last name in this article, despite the disclaimer at the top that "van" is part of the last name... 2607:FEA8:2C5C:3E00:94BA:C5C7:817:EA46 (talk) 13:06, 26 June 2024 (UTC)

A discovery?
I'm wondering why this is under "Discoveries""

"In 1702 he requested a book on Peruvian silver mines in Potosí."

The whole "Discoveries" section oddly written and needs cleaning. 2600:1700:5DD:310:15DD:465:C605:4E4E (talk) 15:40, 19 June 2024 (UTC)

title: Talk:Antonie van Leeuwenhoek


text:
Talk:Environmental Modification Convention

Untitled
Describe the new page here.

The number of states parties is inaccurately listed as 66 according to the US Department of State, which lists 68 state parties, including Brunei (1 Jan 1964) and St Kitts/Nevis (19 Sep 1983).

Cordially,

Prof B H Weston, College of Law, The University of Iowa, Iowa City, IA 52242 USA —Preceding unsigned comment added by 64.6.87.102 (talk • contribs) 26 February 2002

Comments
What is this about? I don't know, and I wondered if the article could explain what it is, not just the countries that have agreed to it. S♦s♦e♦b♦a♦l♦l♦o♦s ( Merry Christmas! ) 03:05, 26 December 2007 (UTC)

External links modified (January 2018)
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Republic of France.
This article may wish elaborate on France's situation as she seems to be entirely out of this treaty, despite being a major nuclear power and a permanent member of the UN Security Council. Thus their abstention is quite unusual and notable. What were her reasons to stay out, why NATO didn't work as a single bloc regarding ENMOD, etc? 158.88.16.4 (talk) 12:10, 2 January 2023 (UTC)


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Gretha Smit

Grietje "Greta" Smit (born 20 January 1976) is a Dutch former speed skater.

Smit won a surprising silver medal in the 2002 Winter Olympics in the 5000 metre event. She skated a world record broken in a later pair by Claudia Pechstein. Prior to that season, she had not competed in long track speed skating competitions for several years, and had only taken up practice again shortly before the Dutch trials in late 2001.

However, she had been a very successful marathon skater in the years before 2002. She had become Dutch marathon skating champion on natural ice in 1995, 1998, 2000 and 2001, and on artificial ice in 1999, 2000 and 2001. During the last held Elfstedentocht in 1997 she finished second. She also won the alternative Elfstedentocht - skated in Finland - in 1998, 2000 and 2001.

Smit's two sisters, Jenita and Marianne, were also good (marathon) skaters; Jenita also took up long track skating in the 2002/2003 season.

title: Gretha Smit


text:
Collaborative software

Collaborative software or groupware is application software designed to help people working on a common task to attain their goals. One of the earliest definitions of groupware is "intentional group processes plus software to support them."

Regarding available interaction, collaborative software may be divided into real-time collaborative editing platforms that allow multiple users to engage in live, simultaneous, and reversible editing of a single file (usually a document); and version control (also known as revision control and source control) platforms, which allow users to make parallel edits to a file, while preserving every saved edit by users as multiple files that are variants of the original file.

Collaborative software is a broad concept that overlaps considerably with computer-supported cooperative work (CSCW). According to Carstensen and Schmidt (1999), groupware is part of CSCW. The authors claim that CSCW, and thereby groupware, addresses "how collaborative activities and their coordination can be supported by means of computer systems."

The use of collaborative software in the work space creates a collaborative working environment (CWE).

Collaborative software relates to the notion of collaborative work systems, which are conceived as any form of human organization that emerges any time that collaboration takes place, whether it is formal or informal, intentional or unintentional. Whereas the groupware or collaborative software pertains to the technological elements of computer-supported cooperative work, collaborative work systems become a useful analytical tool to understand the behavioral and organizational variables that are associated to the broader concept of CSCW.

History
Douglas Engelbart first envisioned collaborative computing in 1951 and documented his vision in 1962, with working prototypes in full operational use by his research team by the mid-1960s. He held the first public demonstration of his work in 1968 in what is now referred to as "The Mother of All Demos". The following year, Engelbart's lab was hooked into the ARPANET, the first computer network, enabling them to extend services to a broader userbase.

Online collaborative gaming software began between early networked computer users. In 1975, Will Crowther created Colossal Cave Adventure on a DEC PDP-10 computer. As internet connections grew, so did the numbers of users and multi-user games. In 1978 Roy Trubshaw, a student at University of Essex in the United Kingdom, created the game MUD (Multi-User Dungeon).

The US Government began using truly collaborative applications in the early 1990s. One of the first robust applications was the Navy's Common Operational Modeling, Planning and Simulation Strategy (COMPASS). The COMPASS system allowed up to 6 users to create point-to-point connections with one another; the collaborative session only remained while at least one user stayed active, and would have to be recreated if all six logged out. MITRE improved on that model by hosting the collaborative session on a server into which each user logged. Called the Collaborative Virtual Workstation (CVW), it allowed the session to be set up in a virtual file cabinet and virtual rooms, and left as a persistent session that could be joined later.

In 1996, Pavel Curtis, who had built MUDs at PARC, created PlaceWare, a server that simulated a one-to-many auditorium, with side chat between "seat-mates", and the ability to invite a limited number of audience members to speak. In 1997, engineers at GTE used the PlaceWare engine in a commercial version of MITRE's CVW, calling it InfoWorkSpace (IWS). In 1998, IWS was chosen as the military standard for the standardized Air Operations Center. The IWS product was sold to General Dynamics and then later to Ezenia.

Groupware
Collaborative software was originally designated as groupware and this term can be traced as far back as the late 1980s, when Richman and Slovak (1987) wrote: "Like an electronic sinew that binds teams together, the new groupware aims to place the computer squarely in the middle of communications among managers, technicians, and anyone else who interacts in groups, revolutionizing the way they work."

In 1978, Peter and Trudy Johnson-Lenz coined the term groupware; their initial 1978 definition of groupware was, "intentional group processes plus software to support them." Later in their article they went on to explain groupware as "computer-mediated culture... an embodiment of social organization in hyperspace." Groupware integrates co-evolving human and tool systems, yet is simply a single system.

In the early 1990s the first commercial groupware products were delivered, and big companies such as Boeing and IBM started using electronic meeting systems for key internal projects. Lotus Notes appeared as a major example of that product category, allowing remote group collaboration when the internet was still in its infancy. Kirkpatrick and Losee (1992) wrote then: "If GROUPWARE really makes a difference in productivity long term, the very definition of an office may change. You will be able to work efficiently as a member of a group wherever you have your computer. As computers become smaller and more powerful, that will mean anywhere." In 1999, Achacoso created and introduced the first wireless groupware.

Design and implementation
The complexity of groupware development is still an issue. One reason is the socio-technical dimension of groupware. Groupware designers do not only have to address technical issues (as in traditional software development) but also consider the organizational aspects and the social group processes that should be supported with the groupware application. Some examples for issues in groupware development are:

 * Persistence is needed in some sessions. Chat and voice communications are routinely non-persistent and evaporate at the end of the session. Virtual room and online file cabinets can persist for years. The designer of the collaborative space needs to consider the information duration needs and implement accordingly.
 * Authentication has always been a problem with groupware. When connections are made point-to-point, or when log-in registration is enforced, it is clear who is engaged in the session. However, audio and unmoderated sessions carry the risk of unannounced 'lurkers' who observe but do not announce themselves or contribute.
 * Until recently, bandwidth issues at fixed location limited full use of the tools. These are exacerbated with mobile devices.
 * Multiple input and output streams bring concurrency issues into the groupware applications.
 * Motivational issues are important, especially in settings without pre-defined group processes in place.
 * Closely related to the motivation aspect is the question of reciprocity. Ellis and others have shown that the distribution of efforts and benefits has to be carefully balanced in order to ensure that all required group members really participate.
 * Real-time communication via groupware can lead to a lot of noise, over-communication, and information overload.

One approach for addressing these issues is the use of design patterns for groupware design. The patterns identify recurring groupware design issues and discuss design choices in a way that all stakeholders can participate in the groupware development process.

Levels of collaboration
Groupware can be divided into three categories depending on the level of collaboration:

 * 1) Communication can be thought of as unstructured interchange of information. A phone call and an instant messaging discussion are examples.
 * 2) Conferencing (or collaboration level, as it is called in academic papers) refers to interactive work toward a shared goal. Brainstorming and voting are examples.
 * 3) Coordination refers to complex interdependent work toward a shared goal. A good metaphor is to think about a sports team; everyone has to contribute the right play at the right time as well as adjust their play to the unfolding situation - but everyone is doing something different - in order for the team to win. It is complex interdependent work toward a shared goal.

Collaborative management (coordination) tools
Collaborative management tools facilitate and manage group activities. Examples include:
 * Document collaboration systems — help people work together on a single document or file to achieve a single final version
 * Electronic calendars (also called time management software) — schedule events and automatically notify and remind group members
 * Project management systems — schedule, track, and chart the steps in a project as it is being completed
 * Online proofing — share, review, approve, and reject web proofs, artwork, photos, or videos between designers, customers, and clients
 * Workflow systems — collaborative management of tasks and documents within a knowledge-based business process
 * Knowledge management systems — collect, organize, manage, and share various forms of information
 * Enterprise bookmarking — collaborative bookmarking engine to tag, organize, share, and search enterprise data
 * Extranet systems (sometimes also known as 'project extranets') — collect, organize, manage, and share information associated with the delivery of a project (e.g., the construction of a building)
 * Intranet systems — quickly share company information via internet to members within a company (e.g., marketing and product info)
 * Social software systems — organize social relations of groups
 * Online spreadsheets — collaborate and share structured data and information
 * Client portals — interact and share with clients in a private online environment

Collaborative software and human interaction
The design intent of collaborative software (groupware) is to transform the way documents and rich media are shared in order to enable more effective team collaboration.

Collaboration, with respect to information technology, seems to have several definitions. Some are defensible but others are so broad they lose any meaningful application. Understanding the differences in human interactions is necessary to ensure the appropriate technologies are employed to meet interaction needs.

There are three primary ways in which humans interact: conversations, transactions, and collaborations.

Conversational interaction is an exchange of information between two or more participants where the primary purpose of the interaction is discovery or relationship building. There is no central entity around which the interaction revolves but is a free exchange of information with no defined constraints, generally focused on personal experiences. Communication technology such as telephones, instant messaging, and e-mail are generally sufficient for conversational interactions.

Transactional interaction involves the exchange of transaction entities where a major function of the transaction entity is to alter the relationship between participants.

In collaborative interaction, the main function of the participants' relationship is to alter a collaboration entity (i.e., the converse of transactional). When teams collaborate on projects it is collaborative project management.

Closely related terms

 * Computer supported cooperative work
 * Integrated collaboration environment

Type of applications

 * Content management system
 * Customer relationship management software
 * Document management system
 * Enterprise content management
 * Intranet

Other related type of applications

 * Massively distributed collaboration
 * Online consultation
 * Online deliberation

Other related terms

 * Cloud collaboration
 * Collaborative innovation network
 * Commons-based peer production
 * Electronic business
 * Information technology management
 * Management information systems
 * Management
 * MediaWiki
 * Office of the future
 * Operational transformation
 * Organizational Memory System
 * Remote work
 * Wikipedia
 * Worknet

title: Collaborative software


text:
User:Darius Bacon

My home page. Github account

title: User:Darius Bacon


text:
Amedeo Avogadro

Lorenzo Romano Amedeo Carlo Avogadro, Count of Quaregna and Cerreto (, also,  ; 9 August 1776 – 9 July 1856) was an Italian scientist, most noted for his contribution to molecular theory now known as Avogadro's law, which states that equal volumes of gases under the same conditions of temperature and pressure will contain equal numbers of molecules. In tribute to him, the ratio of the number of elementary entities (atoms, molecules, ions or other particles) in a substance to its amount of substance (the latter having the unit mole),, is known as the Avogadro constant. This constant is denoted NA, and is one of the seven defining constants of the SI.

Biography

Amedeo Avogadro was born in Turin to a noble family of the Kingdom of Sardinia (now part of Italy) in the year 1776. He graduated in ecclesiastical law at the late age of 20 and began to practice. Soon after, he dedicated himself to physics and mathematics (then called positive philosophy), and in 1809 started teaching them at a liceo (high school) in Vercelli, where his family lived and had some property.

In 1811, he published an article with the title Essai d'une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons ("Essay on a manner of Determining the Relative Masses of the Elementary Molecules of Bodies and the Proportions by Which They Enter These Combinations"), which contains Avogadro's hypothesis. Avogadro submitted this essay to Jean-Claude Delamétherie's Journal de Physique, de Chimie et d'Histoire naturelle ("Journal of Physics, Chemistry and Natural History").

In 1820, he became a professor of physics at the University of Turin. Turin was now the capital of the restored Savoyard Kingdom of Sardinia under Victor Emmanuel I. Avogadro was active in the revolutionary movement of March 1821. As a result, he lost his chair in 1823 (or, as the university officially declared, it was "very glad to allow this interesting scientist to take a rest from heavy teaching duties, in order to be able to give better attention to his researches"). Eventually, King Charles Albert granted a Constitution (Statuto Albertino) in 1848. Well before this, Avogadro had been recalled to the university in Turin in 1833, where he taught for another twenty years.

Little is known about Avogadro's private life, which appears to have been sober and religious. He married Felicita Mazzé and had six children. Avogadro held posts dealing with statistics, meteorology, and weights and measures (he introduced the metric system into Piedmont) and was a member of the Royal Superior Council on Public Instruction.

He died on 9 July 1856.

Accomplishments
In honour of Avogadro's contributions to molecular theory, the number of molecules per mole of a substance is named the Avogadro constant, NA. It is exactly  The Avogadro constant is used to compute the results of chemical reactions. It allows chemists to determine the amounts of substances produced in a given reaction to a great degree of accuracy.

Johann Josef Loschmidt first calculated the value of the Avogadro constant, the number of particles in one mole, sometimes referred to as the Loschmidt number in German-speaking countries (Loschmidt constant now has another meaning).

Avogadro's law states that the relationship between the masses of the same volume of all gases (at the same temperature and pressure) corresponds to the relationship between their respective molecular weights. Hence, the relative molecular mass of a gas can be calculated from the mass of a sample of known volume.

Avogadro developed this hypothesis after Joseph Louis Gay-Lussac published his law on volumes (and combining gases) in 1808. The greatest problem Avogadro had to resolve was the confusion at that time regarding atoms and molecules. One of his most important contributions was clearly distinguishing one from the other, stating that gases are composed of molecules, and these molecules are composed of atoms. (For instance, John Dalton did not consider this possibility.) Avogadro did not actually use the word "atom" as the words "atom" and "molecule" were used almost without difference. He believed that there were three kinds of "molecules", including an "elementary molecule" (our "atom"). Also, he gave more attention to the definition of mass, as distinguished from weight.

In 1815, he published Mémoire sur les masses relatives des molécules des corps simples, ou densités présumées de leur gaz, et sur la constitution de quelques-uns de leur composés, pour servir de suite à l'Essai sur le même sujet, publié dans le Journal de Physique, juillet 1811 ("Note on the Relative Masses of Elementary Molecules, or Suggested Densities of Their Gases, and on the Constituents of Some of Their Compounds, As a Follow-up to the Essay on the Same Subject, Published in the Journal of Physics, July 1811") about gas densities.

In 1821 he published another paper, Nouvelles considérations sur la théorie des proportions déterminées dans les combinaisons, et sur la détermination des masses des molécules des corps (New Considerations on the Theory of Proportions Determined in Combinations, and on Determination of the Masses of Atoms) and shortly afterwards, Mémoire sur la manière de ramener les composès organiques aux lois ordinaires des proportions déterminées ("Note on the Manner of Finding the Organic Composition by the Ordinary Laws of Determined Proportions").

In 1841, he published his work in Fisica dei corpi ponderabili, ossia Trattato della costituzione materiale de' corpi, 4 volumes.

Response to the theory
The scientific community did not give great attention to Avogadro's theory, and it was not immediately accepted. André-Marie Ampère proposed a very similar theory three years later (in his Sur la détermination des proportions dans lesquelles les corps se combinent d'après le nombre et la disposition respective des molécules dont leurs particules intégrantes sont composées; "On the Determination of Proportions in which Bodies Combine According to the Number and the Respective Disposition of the Molecules by Which Their Integral Particles Are Made"), but the same indifference was shown to his theory as well.

Only through studies by Charles Frédéric Gerhardt and Auguste Laurent on organic chemistry was it possible to demonstrate that Avogadro's law explained why the same quantities of molecules in a gas have the same volume.

Unfortunately, related experiments with some inorganic substances showed seeming contradictions. This was finally resolved by Stanislao Cannizzaro, as announced at Karlsruhe Congress in 1860, four years after Avogadro's death. He explained that these exceptions were due to molecular dissociations at certain temperatures, and that Avogadro's law determined not only molecular masses but atomic masses as well.

In 1911, a meeting in Turin commemorated the hundredth anniversary of the publication of Avogadro's classic 1811 paper. King Victor Emmanuel III attended, and Avogadro's great contribution to chemistry was recognized.

Rudolf Clausius, with his kinetic theory on gases proposed in 1857, provided further evidence for Avogadro's law. Jacobus Henricus van 't Hoff showed that Avogadro's theory also held in dilute solutions.

Avogadro is hailed as a founder of the atomic-molecular theory.

title: Amedeo Avogadro