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Evaporation

Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. A high concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidity affects rate of evaporation of water. When the molecules of the liquid collide, they transfer energy to each other based on how they collide. When a molecule near the surface absorbs enough energy to overcome the vapor pressure, it will escape and enter the surrounding air as a gas. When evaporation occurs, the energy removed from the vaporized liquid will reduce the temperature of the liquid, resulting in evaporative cooling.

On average, only a fraction of the molecules in a liquid have enough heat energy to escape from the liquid. The evaporation will continue until an equilibrium is reached when the evaporation of the liquid is equal to its condensation. In an enclosed environment, a liquid will evaporate until the surrounding air is saturated.

Evaporation is an essential part of the water cycle. The sun (solar energy) drives evaporation of water from oceans, lakes, moisture in the soil, and other sources of water. In hydrology, evaporation and transpiration (which involves evaporation within plant stomata) are collectively termed evapotranspiration. Evaporation of water occurs when the surface of the liquid is exposed, allowing molecules to escape and form water vapor; this vapor can then rise up and form clouds. With sufficient energy, the liquid will turn into vapor.

Theory
For molecules of a liquid to evaporate, they must be located near the surface, they have to be moving in the proper direction, and have sufficient kinetic energy to overcome liquid-phase intermolecular forces. When only a small proportion of the molecules meet these criteria, the rate of evaporation is low. Since the kinetic energy of a molecule is proportional to its temperature, evaporation proceeds more quickly at higher temperatures. As the faster-moving molecules escape, the remaining molecules have lower average kinetic energy, and the temperature of the liquid decreases. This phenomenon is also called evaporative cooling. This is why evaporating sweat cools the human body. Evaporation also tends to proceed more quickly with higher flow rates between the gaseous and liquid phase and in liquids with higher vapor pressure. For example, laundry on a clothes line will dry (by evaporation) more rapidly on a windy day than on a still day. Three key parts to evaporation are heat, atmospheric pressure (determines the percent humidity), and air movement.

On a molecular level, there is no strict boundary between the liquid state and the vapor state. Instead, there is a Knudsen layer, where the phase is undetermined. Because this layer is only a few molecules thick, at a macroscopic scale a clear phase transition interface cannot be seen.

Liquids that do not evaporate visibly at a given temperature in a given gas (e.g., cooking oil at room temperature) have molecules that do not tend to transfer energy to each other in a pattern sufficient to frequently give a molecule the heat energy necessary to turn into vapor. However, these liquids are evaporating. It is just that the process is much slower and thus significantly less visible.

Evaporative equilibrium

If evaporation takes place in an enclosed area, the escaping molecules accumulate as a vapor above the liquid. Many of the molecules return to the liquid, with returning molecules becoming more frequent as the density and pressure of the vapor increases. When the process of escape and return reaches an equilibrium, the vapor is said to be "saturated", and no further change in either vapor pressure and density or liquid temperature will occur. For a system consisting of vapor and liquid of a pure substance, this equilibrium state is directly related to the vapor pressure of the substance, as given by the Clausius–Clapeyron relation:

 * $$\ln \left( \frac{ P_2 }{ P_1 } \right) = - \frac{ \Delta H_{\rm vap } }{ R } \left( \frac{ 1 }{ T_2 } - \frac{ 1 }{ T_1 }  \right)$$

where P1, P2 are the vapor pressures at temperatures T1, T2 respectively, ΔHvap is the enthalpy of vaporization, and R is the universal gas constant. The rate of evaporation in an open system is related to the vapor pressure found in a closed system. If a liquid is heated, when the vapor pressure reaches the ambient pressure the liquid will boil.

The ability for a molecule of a liquid to evaporate is based largely on the amount of kinetic energy an individual particle may possess. Even at lower temperatures, individual molecules of a liquid can evaporate if they have more than the minimum amount of kinetic energy required for vaporization.

Factors influencing the rate of evaporation
Note: Air is used here as a common example of the surrounding gas; however, other gases may hold that role.

 * Concentration of the substance evaporating in the air: If the air already has a high concentration of the substance evaporating, then the given substance will evaporate more slowly.
 * Flow rate of air: This is in part related to the concentration points above. If "fresh" air (i.e., air which is neither already saturated with the substance nor with other substances) is moving over the substance all the time, then the concentration of the substance in the air is less likely to go up with time, thus encouraging faster evaporation. This is the result of the boundary layer at the evaporation surface decreasing with flow velocity, decreasing the diffusion distance in the stagnant layer.

 * The amount of minerals dissolved in the liquid

 * Inter-molecular forces: The stronger the forces keeping the molecules together in the liquid state, the more energy one must get to escape. This is characterized by the enthalpy of vaporization.

 * Pressure: Evaporation happens faster if there is less exertion on the surface keeping the molecules from launching themselves.

 * Surface area: A substance that has a larger surface area will evaporate faster, as there are more surface molecules per unit of volume that are potentially able to escape.

 * Temperature of the substance: the higher the temperature of the substance the greater the kinetic energy of the molecules at its surface and therefore the faster the rate of their evaporation.

 * Photomolecular effect: The amount of light will affect the evaporation. When photons hits the surface area of the liquid they can make individual molecules break free and disappear into the air without any need for additional heat.

In the US, the National Weather Service measures, at various outdoor locations nationwide, the actual rate of evaporation from a standardized "pan" open water surface. Others do likewise around the world. The US data is collected and compiled into an annual evaporation map. The measurements range from under 30 to over 120 in per year.

Because it typically takes place in a complex environment, where 'evaporation is an extremely rare event', the mechanism for the evaporation of water is not completely understood. Theoretical calculations require prohibitively long and large computer simulations. 'The rate of evaporation of liquid water is one of the principal uncertainties in modern climate modeling.'

Thermodynamics
Evaporation is an endothermic process, since heat is absorbed during evaporation.

Applications

 * Industrial applications include many printing and coating processes; recovering salts from solutions; and drying a variety of materials such as lumber, paper, cloth and chemicals.
 * The use of evaporation to dry or concentrate samples is a common preparatory step for many laboratory analyses such as spectroscopy and chromatography. Systems used for this purpose include rotary evaporators and centrifugal evaporators.
 * When clothes are hung on a laundry line, even though the ambient temperature is below the boiling point of water, water evaporates. This is accelerated by factors such as low humidity, heat (from the sun), and wind. In a clothes dryer, hot air is blown through the clothes, allowing water to evaporate very rapidly.
 * The matki/matka, a traditional Indian porous clay container used for storing and cooling water and other liquids.
 * The botijo, a traditional Spanish porous clay container designed to cool the contained water by evaporation.
 * Evaporative coolers, which can significantly cool a building by simply blowing dry air over a filter saturated with water.

Combustion vaporization
Fuel droplets vaporize as they receive heat by mixing with the hot gases in the combustion chamber. Heat (energy) can also be received by radiation from any hot refractory wall of the combustion chamber.

Pre-combustion vaporization
Internal combustion engines rely upon the vaporization of the fuel in the cylinders to form a fuel/air mixture in order to burn well. The chemically correct air/fuel mixture for total burning of gasoline has been determined to be 15 parts air to one part gasoline or 15/1 by weight. Changing this to a volume ratio yields 8000 parts air to one part gasoline or 8,000/1 by volume.

Film deposition
Thin films may be deposited by evaporating a substance and condensing it onto a substrate, or by dissolving the substance in a solvent, spreading the resulting solution thinly over a substrate, and evaporating the solvent. The Hertz–Knudsen equation is often used to estimate the rate of evaporation in these instances.

title: Evaporation


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Esbat

An esbat is a coven meeting or ritual at a time other than one of the Sabbats within Wicca and other Wiccan-influenced forms of contemporary Paganism.

Esbats can span a wide range of purposes from coven business meetings and initiation ceremonies to social gatherings, times of merriment, and opportunities to commune with the divine. Janet and Stewart Farrar describe esbats as an opportunity for a "love feast, healing work, psychic training and all."

Esbats are typically held once per month on or near the night of a full moon or new moon. Due to the connection between the Moon and femininity in Wicca, esbats are associated with goddesses.

Esbats are a time set aside for formal worship and have been described as similar to Sundays for Christians or Friday nights for Jewish people. They can be solitary affairs but tend to be conducted in groups. Sources vary on whether these rituals are open to the public or only to initiated members.

Etymology
The term esbat is derived from Old French s'esbattre (Modern French ébat), meaning to frolic and amuse oneself, diversion. It was a borrowing by 20th century anthropologist Margaret Murray's use of French witch trial sources on supposed Witches' Sabbaths in her attempts to "reconstruct" a Witch Cult in Western Europe.

Observance
An esbat is commonly understood to be a ritual observance on the night of a full moon. However, the late high priestess Doreen Valiente distinguished between "full moon Esbat[s]" and other esbatic occasions.

The term esbat in this sense was described by Margaret Murray. "The Esbat differed from the Sabbat by being primarily for business. ... very often the Esbat was for sheer enjoyment only"Esbats vary greatly and can be simple or elaborate. Rituals use symbolism to enhance the properties of a particular moon, of which there are 13 per solar year.

They are typically held at coven members' homes or outdoors. Tools such as candles, athames, incense, pentacles, items from nature, bowls of water, mirrors, and crystals are commonly placed on an altar. The ceremony begins with participants establishing a sacred space by casting a circle or purifying the area by smudging. Then, they commune with the divine, pray, and meditate. Central elements are reflection on changes that have occurred in the past moon cycle, things you wish to change by the next moon, and gratitude. Once the ceremony is completed, a ritual closing is performed. Participants describe esbats as "spiritually fulfilling" and "immensely beneficial to our personal spiritual growth."

New Moon
When esbat rituals occur during a new moon (also known as a dark moon), it is an occasion to worship the darker aspects of witchcraft. This represents elements that are hidden or in shadow, and is not necessarily associated with evil. New moon esbats may be used to worship a dark or maiden goddess, to banish something unwanted, or to end a phase in life. Because esbats typically occur when the moon is at its highest point, covens may choose for new moon esbats to take place in the mid-afternoon.

Full Moon
The full moon esbat tends to be a frenetic celebration. Spells for wholeness, children, mothers, families, clairvoyance, and love are performed. Most full moon esbats are held at midnight because the moon is most visible which allows participants to feel closer to it.

One major component of full moon esbats is drawing down the moon. The idea is to draw the essence of the lunar goddess into the body of a coven member, usually a priestess or leader. This is done by another member who channels lunar energy down through a receptacle such as a chalice or knife. The two stand facing each other in the center of the circle and the group asks the goddess to come down. Once the divine energy enters the tool, it is touched to the head, chest, or abdomen of the priestess. Through her body, the goddess speaks to the group by answering questions, giving instructions, offering blessings, or simply "[pouring] her loving energy into the circle and [leading] you in a merry spiral dance." After returning the energy to the goddess, the sign of a pentagram may be used as a dismissal.

The ceremony of cakes and ale is the other main component which typically concludes the full moon esbat. This ritual uses cookies or a loaf of bread and a chalice of wine or other drink which are placed on a central altar. The bread and wine symbolize the body and blood of the mother goddess, who gives life to all things. Each participant takes turns blessing the bread, breaking off a piece, eating it, then passing it clockwise. The same process happens with the chalice. This allows everyone to honor the body and blood of the goddess as they wish.

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

title: Esbat


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Equal temperament

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.

In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12 tone equal temperament, 12 or 12, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2, ( $\sqrt{2|12}$ ≈ 1.05946 ). That resulting smallest interval, $1⁄12$ the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12.

In modern times, 12 is usually tuned relative to a standard pitch of 440 Hz, called A 440, meaning one note, A (musical note), is tuned to 440 hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.

Other equal temperaments divide the octave differently. For example, some music has been written in 19 and 31, while the Arab tone system uses 24.

Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts.

For tuning systems that divide the octave equally, but are not approximations of just intervals, the term equal division of the octave, or  can be used.

Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.

General properties
In an equal temperament, the distance between two adjacent steps of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly spaced and would not permit transposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio:

 * $$\ r^n = p\ $$
 * $$\ r = \sqrt[n]{p\ }\ $$

where the ratio $3⁄2$ divides the ratio $5⁄4$ (typically the octave, which is 2:1) into $7⁄4$ equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in ethnomusicology. The basic step in cents for any equal temperament can be found by taking the width of $11⁄8$ above in cents (usually the octave, which is 1200 cents wide), called below $13⁄8$, and dividing it into $r$ parts:
 * $$\ c = \frac{\ w\ }{ n }\ $$

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., $p$ is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

Twelve-tone equal temperament
12 tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music.

History
The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu, it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later."

The developments occurred independently.

Kenneth Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 1584 Zhu wrote:
 * I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations.

Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications". Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.

China
Chinese theorists had previously come up with approximations for 12, but Zhu was the first person to mathematically solve 12 tone equal temperament, which he described in two books, published in 1580 and 1584. Needham also gives an extended account.

Zhu obtained his result by dividing the length of string and pipe successively by $\sqrt{2$, and for pipe length by $\sqrt{2$ , such that after 12 divisions (an octave), the length was halved.

Zhu created several instruments tuned to his system, including bamboo pipes.

Europe
Some of the first Europeans to advocate equal temperament were lutenists Vincenzo Galilei, Giacomo Gorzanis, and Francesco Spinacino, all of whom wrote music in it.

Simon Stevin was the first to develop 12 based on the twelfth root of two, which he described in van de Spiegheling der singconst (c. 1605), published posthumously in 1884.

Plucked instrument players (lutenists and guitarists) generally favored equal temperament, while others were more divided. In the end, 12-tone equal temperament won out. This allowed enharmonic modulation, new styles of symmetrical tonality and polytonality, atonal music such as that written with the 12-tone technique or serialism, and jazz (at least its piano component) to develop and flourish.

Mathematics

In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

 * $$ \sqrt[12]{2\ } = 2^{\tfrac{1}{12}} \approx 1.059463 $$

This interval is divided into 100 cents.

Calculating absolute frequencies
To find the frequency, $Pn$, of a note in 12, the following formula may be used:

 * $$\ P_n = P_a\ \cdot\ \Bigl(\ \sqrt[12]{2\ }\ \Bigr)^{ n-a }\ $$

In this formula $Pn$ represents the pitch, or frequency (usually in hertz), you are trying to find. $Pa$ is the frequency of a reference pitch. The indes numbers $n$ and $p$ are the labels assigned to the desired pitch ($w$) and the reference pitch ($n$). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, $c$ (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and $vii$ (middle ), and ♯$n$ are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of $a$ and ♯$n$:

 * $$P_{40} = 440\ \mathsf{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(40-49)} \approx 261.626\ \mathsf{Hz}\ $$
 * $$P_{46} = 440\ \mathsf{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(46-49)} \approx 369.994\ \mathsf{Hz}\ $$

Converting frequencies to their equal temperament counterparts
To convert a frequency (in Hz) to its equal 12 counterpart, the following formula can be used:

 * $$\ E_n = E_a\ \cdot\ 2^{\ x}\ \quad $$ where in general $$ \quad\ x\ \equiv\ \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12\log_{2} \left(\frac{\ n\ }{ a }\right) \Biggr) ~.$$

$En$ is the frequency of a pitch in equal temperament, and $Ea$ is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that $a$ and ♯$4$ have the following frequencies, respectively:

 * $$E_{660} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 7 }{\ 12\ }\right)}\ \approx\ 659.255\ \mathsf{Hz}\ \quad $$ where in this case $$ \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl(\ 12 \log_{2}\left(\frac{\ 660\ }{ 440 }\right)\ \Biggr) = \frac{ 7 }{\ 12\ } ~.$$

 * $$E_{550} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 1 }{\ 3\ }\right)}\ \approx\ 554.365\ \mathsf{Hz}\ \quad $$ where in this case $$ \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12 \log_{2}\left(\frac{\ 550\ }{ 440 }\right)\Biggr) = \frac{ 4 }{\ 12\ } = \frac{ 1 }{\ 3\ } ~.$$

Comparison with just intonation
The intervals of 12 closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.

In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.

 * {| class="wikitable" style="margin:auto;text-align:center;"

! Interval Name ! Exact value in 12 ! Decimal value in 12 ! Pitch in ! Just intonation interval ! Cents in just intonation ! 12 cents tuning error
 * Unison (C (musical note))
 * 2$4$ = 1
 * 1
 * 0
 * $4$ = 1
 * 0
 * 0
 * Minor second (♭)
 * 2$4$ = $4$
 * 100
 * $5$ = 1.06666...
 * Major second (D (musical note))
 * 2$5$ = $0/12$
 * 200
 * $1⁄1$ = 1.125
 * Minor third (♭)
 * 2$1/12$ = $\sqrt{{{big|2}}|12}$
 * 300
 * $16⁄15$ = 1.2
 * Major third (E (musical note))
 * 2$2/12$ = $\sqrt{{{big|2}}|6}$
 * 400
 * $9⁄8$ = 1.25
 * Perfect fourth (F (musical note))
 * 2$3/12$ = $\sqrt{{{big|2}}|4}$
 * 500
 * $6⁄5$ = 1.33333...
 * Tritone (♭)
 * 2$4/12$ = √2
 * 600
 * $\sqrt{{{big|2}}|3}$= 1.42222...
 * Perfect fifth (G (musical note))
 * 2$5⁄4$ = $5/12$
 * 700
 * $\sqrt{{{big|32}}|12}$ = 1.5
 * Minor sixth (♭)
 * 2$4⁄3$ = $6/12$
 * 800
 * $64⁄45$ = 1.6
 * Major sixth (A (musical note))
 * 2$7/12$ = $\sqrt{{{big|128}}|12}$
 * 900
 * $3⁄2$ = 1.66666...
 * Minor seventh (♭)
 * 2$8/12$ = $\sqrt{{{big|4}}|3}$
 * 1000
 * $8⁄5$ = 1.77777...
 * Major seventh (B (musical note))
 * 2$9/12$ = $\sqrt{{{big|8}}|4}$
 * 1100
 * $5⁄3$ = 1.875
 * 0
 * Octave (C (musical note))
 * 2$10/12$ = 2
 * 2
 * 1200
 * $\sqrt{{{big|32}}|6}$ = 2
 * 1200.00
 * 0
 * }
 * 700
 * $16⁄9$ = 1.5
 * Minor sixth (♭)
 * 2$11/12$ = $\sqrt{{{big|2048}}|12}$
 * 800
 * $15⁄8$ = 1.6
 * Major sixth (A (musical note))
 * 2$12/12$ = $2⁄1$
 * 900
 * $\sqrt{{{small|{{sfrac|3|2}}}}|7}$ = 1.66666...
 * Minor seventh (♭)
 * 2$1⁄3$ = $2$
 * 1000
 * $\sqrt{3|13}$ = 1.77777...
 * Major seventh (B (musical note))
 * 2$\sqrt{{{small|{{sfrac|3|2}}}}|9}$ = $\sqrt{{{small|{{sfrac|3|2}}}}|11}$
 * 1100
 * $\sqrt{{{small|{{sfrac|3|2}}}}|20}$ = 1.875
 * 0
 * Octave (C (musical note))
 * 2$s$ = 2
 * 2
 * 1200
 * $t$ = 2
 * 1200.00
 * 0
 * }
 * Major seventh (B (musical note))
 * 2$q$ = $k$
 * 1100
 * $k$ = 1.875
 * 0
 * Octave (C (musical note))
 * 2$k$ = 2
 * 2
 * 1200
 * $k$ = 2
 * 1200.00
 * 0
 * }
 * 1200
 * $k$ = 2
 * 1200.00
 * 0
 * }

Seven-tone equal division of the fifth
Violins, violas, and cellos are tuned in perfect fifths ( for violins and for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12 tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of $k$ to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.

Five-, seven-, and nine-tone temperaments in ethnomusicology

Five- and seven-tone equal temperament (5 and  ), with 240 cent  and 171 cent  steps, respectively, are fairly common.

5 and 7 mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1.
 * In 5, the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents.
 * In 7, the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171 cents each).

5 tone and 9 tone equal temperament
According to Kunst (1949), Indonesian gamelans are tuned to 5, but according to Hood (1966) and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretched octaves. It is now accepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to 9-TET (133-cent steps ).

7-tone equal temperament
A Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from 7. According to Morton,
 * "Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode."

A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.

Chinese music has traditionally used 7.

Various equal temperaments

 * 19 EDO: Many instruments have been built using 19 EDO tuning. Equivalent to $n$ comma meantone, it has a slightly flatter perfect fifth (at 695 cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19 EDO being 232 EDO. Its perfect fourth (at 505 cents), is seven cents sharper than just intonation's and five cents sharper than 12 EDO's.

 * 22 EDO: 22 EDO is the smallest EDO to represent superpyth temperament (when 7:4 and 16:9 are the same interval) and is near the optimal generator for porcupine temperament. The fifths are so sharp that the major and minor intervals we get from stacking fifths will be the same as supermajor and subminor versions of those intervals. One step closer to each other are the more "in tune" versions of major and minor thirds (5/4 and 6/5).

 * 23 EDO: 23 EDO is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents. But it does approximate ratios between them (including the justly-tuned 6/5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory.

 * 24 EDO: 24 EDO, the quarter-tone scale, is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who are also interested in microtonality. Because 24 EDO contains all the pitches of 12 EDO, musicians employ the additional colors without losing any tactics available in 12 tone harmony. That 24 is a multiple of 12 also makes 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12 tone notation. Various composers, including Charles Ives, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO.

 * 26 EDO: 26 is the lowest number of equal divisions of the octave that almost purely tunes the 7th harmonic (7:4). Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a neutral third rather than a major third. 26 EDO has two minor thirds and two minor sixths and could be an alternate temperament for barbershop harmony.

 * 27 EDO: 27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the septimal comma but not the syntonic comma.

 * 29 EDO: 29 is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12 EDO, in which the fifth is 1.5 cents sharp instead of 2 cents flat. Its classic major third is roughly as inaccurate as 12 EDO, but is tuned 14 cents flat rather than 14 cents sharp. It also tunes the 7th, 11th, and 13th harmonics flat, by roughly the same amount, allowing 29 EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces 58 EDO, which allows for lower errors for some just tones.

 * 31 EDO: 31 EDO was advocated by Christiaan Huygens and Adriaan Fokker and represents a standardization of quarter-comma meantone. 31 EDO does not have as accurate a perfect fifth as 12 EDO (like 19 EDO), but its major thirds and minor sixths are less than 1 cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate.

 * 34 EDO: 34 EDO gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31 EDO does, despite having a slightly less accurate fit for 5:4. 34 EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is sharp instead of flat. It enables the 600 cent tritone, since 34 is an even number.

 * 41 EDO: 41 is the second-lowest number of equal divisions of the octave with a better perfect fifth than 12 EDO. Its classic major third is more accurate than 12 EDO and 29 EDO, at six cents flat. It is not meantone, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31 EDO. It is more accurate in the 13 limit than 31 EDO.

 * 46 EDO: 46 EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many say that this gives major triads a characteristic bright sound. The harmonics up to 11 are within 5 cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a meantone system, it distinguishes 10:9 and 9:8.

 * 53 EDO: 53 EDO has only had occasional use, but is better at approximating the traditional just consonances than 12, 19 or 31 EDO. Its extremely accurate perfect fifths make it equivalent to an extended Pythagorean tuning, and it is sometimes used in Turkish music theory. It does not, however, fit the technical requirements of meantone temperaments, which put good thirds within easy reach, via the cycle of fifths. In 53 EDO, the very consonant thirds are instead reached by using a Pythagorean diminished fourth (C-F♭), as it is an example of schismatic temperament, like 41 EDO.

 * 58 EDO: 58 equal temperament is a duplication of 29 EDO, which it contains as an embedded temperament. Like 29 EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and sixths.

 * 72 EDO: 72 EDO approximates many just intonation intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72 EDO has been taught, written and performed in practice by Joe Maneri and his students (whose atonal inclinations typically avoid any reference to just intonation whatsoever). It can be considered an extension of 12 EDO because 72 is a multiple of 12. 72 EDO does not accurately approximate the 13th harmonic or most simple ratios involving 13. It contains six copies of 12 EDO starting on different pitches, three copies of 24 EDO, and two copies of 36 EDO, which are themselves multiples of 12.

 * 96 EDO: 96 EDO approximates all intervals within 6.25 cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12 EDO. It has been advocated by several composers, especially Julián Carrillo.

Other equal divisions of the octave that have found occasional use include 13 EDO, 15 EDO and 17 EDO.

2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of log$1⁄3$(3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601 just twelfths/fifths than in any equal temperament with fewer tones.

1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.

Equal temperaments of non-octave intervals
The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1 (1902 cents) conventionally a perfect fifth plus an octave (that is, a perfect twelfth), called in this theory a tritave, and split into 13 equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents, or $2⁄3$.

Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were called alpha, beta, and gamma. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes: Alpha and beta may be heard on the title track of Carlos's 1986 album Beauty in the Beast.
 * alpha: $2⁄5$ (78.0 cents)
 * beta: $3⁄5$ (63.8 cents)
 * gamma: $k$ (35.1 cents)

Proportions between semitone and whole tone
In this section, semitone and whole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be $q$, and the number of steps in a tone be $1⁄7$.

There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example,, , , , and ♯ are in ascending order if they preserve their usual relationships to ). That is, fixing $8⁄9$ to a proper fraction in the relationship $q t = s$ also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where $1⁄3$ is an integer, 12$2⁄3$ sets $q = 1⁄2$, 19 $T t s T t T s$ sets $q = 1⁄3$, and 31 $s$ sets $q = 2⁄5$. The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24, the half-sharps and half-flats are not in the circle of fifths generated starting from .) The extreme cases are 5 $T$ , where $q = 0$ and the semitone becomes a unison, and 7 $t$, where $q = 1$ and the semitone and tone are the same interval.

Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into $7 t − 2 s$ steps and the perfect fifth into $4 t − s$ steps. If there are notes outside the circle of fifths, one must then multiply these results by $κ$, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24, six in 72 ). (One must take the small semitone for this purpose: 19 has two semitones, one being $T T t s$ tone and the other being $T t t s$. Similarly, 31  has two semitones, one being $κ$ tone and the other being $t = s c$).

The smallest of these families is 12 $T = s c κ$, and in particular, 12 is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12 has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.)

Each choice of fraction $T t s T t T s$ for the relationship results in exactly one equal temperament family, but the converse is not true: 47 has two different semitones, where one is $s c κ   s c   s   s c κ   s c   s c κ   s$ tone and the other is $s$, which are not complements of each other like in 19  ($c$ and $κ$). Taking each semitone results in a different choice of perfect fifth.

Regular diatonic tunings
The diatonic tuning in 12 tone equal temperament (12 ) can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps $s$ (or some circular shift or "rotation" of it). To be called a regular diatonic tuning, each of the two semitones ($c$) must be smaller than either of the tones (greater tone, $s$, and lesser tone, $c$). The comma $s$ is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies $κ = T⁄t$, or as cents $κ = T − t$.

The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does not close (unlike the circle of fifths in 12 ). Starting on the subdominant (in the key of C) there are three perfect fifths in a row—–, –, and –—each a composite of some permutation of the smaller intervals $c$ . The three in-tune fifths are interrupted by the grave fifth – = $κ$(grave means "flat by a comma"), followed by another perfect fifth, –, and another grave fifth, –♯, and then restarting in the sharps with ♯–♯; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by fourths the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are "wolf" intervals: Each of the grave fifths out of tune by a diatonic comma.

Since the comma, $T$, expands the lesser tone  $t$, into the greater tone,  $s$ , a just octave $c$ can be broken up into a sequence  $s$ , (or a circular shift of it) of diatonic semitones $T$, chromatic semitones $t$, and commas  $c$. Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones $κ$, or into the five chromatic semitones $s$, or into both $κ$ and $s = c$, with some fixed proportion for each type of semitone.

The sequence of intervals $s$, $κ$, and $t t t t t$ can be repeatedly appended to itself into a greater spiral of 12 fifths, and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.

Morphing diatonic tunings into EDO
An equal temperament can be created if the sizes of the major and minor tones ($s$, $κ$) are altered to be the same (say, by setting $κ = 0$, with the others expanded to still fill out the octave), and both semitones ($t t t t t t t$ and $s = 2 c$) the same size, then twelve equal semitones, two per tone, result. In 12 |undefined, the semitone, $c$, is exactly half the size of the same-size whole tones $s$ = $T$.

Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains 7 in the limit as the size of $t$ and $1⁄3$ tend to zero, with the octave kept fixed, and 5 in the limit as $1⁄4$ and $1⁄5$ tend to zero; 12 is of course, the case  ⇭⇭⇭  and  $κ = 0$. For instance:

 * 5 and 7: There are two extreme cases that bracket this framework: When ⇭⇭⇭ and ⇭⇭⇭ reduce to zero with the octave size kept fixed, the result is ⇭⇭⇭, a 5 tone equal temperament. As the ⇭⇭⇭ gets larger (and absorbs the space formerly used for the comma ⇭⇭⇭), eventually the steps are all the same size, ⇭⇭⇭ , and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.

 * 19: If the diatonic semitone is set double the size of the chromatic semitone, i.e.  ⇭⇭⇭ (in cents) and  $κ = 0$, the result is 19 , with one step for the chromatic semitone ⇭⇭⇭, two steps for the diatonic semitone ⇭⇭⇭, three steps for the tones ⇭⇭⇭ = ⇭⇭⇭, and the total number of steps $ 3 T + 2 t + 2 s = 9 + 6 + 4 = $ 19 steps. The imbedded 12 tone sub-system closely approximates the historically important ⇭⇭⇭ comma meantone system.

 * 31: If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. $c = 2⁄3 s$, with $κ = 0$ , the result is 31 , with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where $ 3 T + 2 t + 2 s = 15 + 10 + 6 = $ 31 steps. The imbedded 12 tone sub-system closely approximates the historically important ⇭⇭⇭ comma meantone.

 * 43: If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e. $c = 3⁄4 s$, with $κ = 0$ , the result is 43 , with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where $ 3 T + 2 t + 2 s = 21 + 14 + 8 = $ 43. The imbedded 12 tone sub-system closely approximates ⇭⇭⇭ comma meantone.

 * 53: If the chromatic semitone is made the same size as three commas,  $c = 3 κ$  (in cents, in frequency  $c = κ³$ ) the diatonic the same as five commas,  $s = 5 κ$, that makes the lesser tone eight commas $t = s + c = 8 κ$ , and the greater tone nine,  $T = s + c + κ = 9 κ$. Hence $ 3 T + 2 t + 2 s = 27 κ + 16 κ + 10 κ = 53 κ$ for 53 steps of one comma each. The comma size / step size is  $κ = 1300⁄53$ ¢ exactly, or  $κ = 22.642$ ¢  $≈ 21.506$ ¢ , the syntonic comma. It is an exceedingly close approximation to 5-limit just intonation and Pythagorean tuning, and is the basis for Turkish music theory.

title: Equal temperament


text:
Talk:Euclidean geometry

Where are the Axioms?
Some people say that Euclid's book don't contain axioms; I have no reason to argue that, not knowing Ancient Greek. But we are in 21st century, and more than one set of axioms of Euclidean Geometry is available, and I am sure must be either listed or refered. Any Mathematicians present here? I could write a section describing an axiomatic Euclidean Geometry, as it is what people expect from a geometry theory. Using a set-theoretical model of geometry is not enough for describing a theory. First, I hope at least some people know that there's more than one set theory; second, we are talking here about a theory, not about models, right?

I'd love to hear your opinion. Vlad Patryshev (talk) 15:29, 14 May 2022 (UTC)

The article mentions Playfair's axiom, perhaps you could develop that. You can also expand the logical basis section Lbertolotti (talk) 23:16, 9 June 2022 (UTC)
 * Some of the best known axiomatic formulations of Euclidean geometry are listed in section . Sure that this section should be made more visible and could be improved, but there is no reason for rewriting it from scratch.
 * You wrote "I could write a section describing an axiomatic Euclidean Geometry". As you do not say which existing axiomatisation you have in mind, I suppose that this would be your own axiomatization. This would be WP:OR, and, as such, would be strictly forbidden in Wikipedia. D.Lazard (talk) 08:58, 10 June 2022 (UTC)

Euclid's Geometry
What are postulates ? 103.170.68.3 (talk) 14:33, 29 June 2022 (UTC)
 * Follow the link: postulate. D.Lazard (talk) 15:27, 29 June 2022 (UTC)

consider this image for your article
https://upload.wikimedia.org/wikipedia/commons/5/59/Picture_euclidean_geometry_123.png Fausto!'20045 (talk) 00:07, 20 April 2023 (UTC)

 * This image does not really seem useful or relevant here. It is a low-resolution screengrab of a Geogebra (?) demonstration that three angles of a triangle (specifically the isosceles triangle with base length 2 and altitude 2) sum to a straight angle. There are many bizarre choices: poorly placed labels, an ugly color scheme, the unexplained inclusion of a Cartesian coordinate grid relative to which the triangle seems to be randomly located, etc. –jacobolus (t) 00:28, 20 April 2023 (UTC)

"An absolute, often metaphysical sense"?
According to the current article:

 * For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century.

I think the main idea being conveyed here is basically correct, but as these sentences stand, they're not very well-written.

I would point out a few problems:

1. The language used here is very strong, probably too strong: "For more than two thousand years...no other sort of geometry had been conceived." If somebody here thinks this language is appropriate, can they cite a source for keeping the current wording? The SEP makes the point that I think is trying to be made here in a more measured way: "The epistemologically convincing status of Euclid’s Elements was uncontested by almost everyone until the later decades of the 19th century."

2. Similarity, "any theorem proved from [Euclid's postulates] was deemed true in an absolute, often metaphysical, sense" is not well-written. Deemed by whom? And what is the "absolute, often metaphysical sense" being spoken of in this sentence? There were lots of competing systems of metaphysics in the ancient world and likewise in the Medieval world. In whose "metaphysical sense" were the theorems of Euclidean geometry deemed absolutely true? Again, the SEP makes the point that I think is trying to be made here, but in language that is a lot clearer: "Euclid’s treatment of geometry has, through the ages, been celebrated as a perfect deductive presentation of a science, and certainly Euclid made great efforts to obtain a most careful logical chain of truths." DefinitelyNotAnExtraterrestrial (talk) 02:57, 20 May 2023 (UTC)

 * . D.Lazard (talk) 10:38, 20 May 2023 (UTC)

Two rotten edit summaries: my apologies
In "System of measurement and arithmetic" I have just changed "congruent" to "equal", and in the Edit Summary I twice tried to say "Hasn't an earlier sentence just said that 'congruence' applies to the entire figure?"

While editing the Edit Summary I have now learned the hard way that pressing Enter does not go to a new line, it IMMEDIATELY SUBMITS the edit, before I was ready.

I am using a tablet and I suppose that mobile editing works differently to what (I hope) the WP creators intended. If anyone with more WP knowledge can get the system changed so it doesn't do that, please do. 203.220.1.139 (talk) 13:55, 30 September 2023 (UTC)

title: Talk:Euclidean geometry


text:
Edward Gibbon

Edward Gibbon (8 May 1737  – 16 January 1794) was an English essayist, historian, and politician. His most important work, The History of the Decline and Fall of the Roman Empire, published in six volumes between 1776 and 1789, is known for the quality and irony of its prose, its use of primary sources, and its polemical criticism of organized religion.

Early life: 1737–1752
Edward Gibbon was born in 1737, the son of Edward and Judith Gibbon, at Lime Grove in the town of Putney, Surrey. He had five brothers and one sister, all of whom died in infancy. His grandfather, also named Edward, had lost his assets as a result of the South Sea bubble stock-market collapse in 1720 but eventually regained much of his wealth. Gibbon's father thus inherited a substantial estate. His paternal grandmother, Catherine Acton, was granddaughter of Sir Walter Acton, 2nd Baronet.

As a youth, Gibbon's health was under constant threat. He described himself as "a puny child, neglected by my Mother, starved by my nurse". At age nine, he was sent to Dr. Woddeson's school at Kingston upon Thames (now Kingston Grammar School), shortly after which his mother died. He then took up residence in the Westminster School boarding house, owned by his adored "Aunt Kitty", Catherine Porten. Soon after she died in 1786, he remembered her as rescuing him from his mother's disdain, and imparting "the first rudiments of knowledge, the first exercise of reason, and a taste for books which is still the pleasure and glory of my life". From 1747 Gibbon spent time at the family home in Buriton. By 1751, Gibbon's reading was already extensive and pointed toward his future pursuits: Laurence Echard's Roman History (1713), William Howel(l)'s An Institution of General History (1680–85), and several of the 65 volumes of the acclaimed Universal History from the Earliest Account of Time (1747–1768).

Oxford, Lausanne, and a religious journey: 1752–1758
Following a stay at Bath in 1752 to improve his health at the age of 15, Gibbon was sent by his father to Magdalen College, Oxford, where he was enrolled as a gentleman-commoner. He was ill-suited, however, to the college atmosphere, and later rued his 14 months there as the "most idle and unprofitable" of his life. Because he says so in his autobiography, it used to be thought that a penchant from his aunt for "theological controversy" bloomed under the influence of the deist or rationalist theologian Conyers Middleton (1683–1750), the author of Free Inquiry into the Miraculous Powers (1749). In that tract, Middleton denied the validity of such powers; Gibbon promptly objected, or so the argument used to run. The product of that disagreement, with some assistance from the work of Catholic Bishop Jacques-Bénigne Bossuet (1627–1704), and that of the Elizabethan Jesuit Robert Parsons (1546–1610), yielded the most memorable event of his time at Oxford: his conversion to Roman Catholicism on 8 June 1753. He was further "corrupted" by the 'free thinking' deism of the playwright and poet David Mallet; and finally Gibbon's father, already "in despair," had had enough. David Womersley has shown, however, that Gibbon's claim to having been converted by a reading of Middleton is very unlikely, and was introduced only into the final draft of the "Memoirs" in 1792–93.

Within weeks of his conversion, he was removed from Oxford and sent to live under the care and tutelage of Daniel Pavillard, Reformed pastor of Lausanne, Switzerland. There, he made one of his life's two great friendships, that of Jacques Georges Deyverdun (the French-language translator of Goethe's The Sorrows of Young Werther), and that of John Baker Holroyd (later Lord Sheffield). Just a year and a half later, after his father threatened to disinherit him, on Christmas Day, 1754, he reconverted to Protestantism. "The various articles of the Romish creed," he wrote, "disappeared like a dream".

Thwarted romance
He also met the one romance in his life: the daughter of the pastor of Crassy, a young woman named Suzanne Curchod, who was later to become the wife of Louis XVI's finance minister Jacques Necker, and the mother of Madame de Staël. The two developed a warm affinity; Gibbon proceeded to propose marriage, but ultimately this was out of the question, blocked both by his father's staunch disapproval and Curchod's equally staunch reluctance to leave Switzerland. Gibbon returned to England in August 1758 to face his father. No refusal of the elder's wishes could be allowed. Gibbon put it this way: "I sighed as a lover, I obeyed as a son." He proceeded to cut off all contact with Curchod, even as she vowed to wait for him. Their final emotional break apparently came at Ferney, France, in early 1764, though they did see each other at least one more time a year later.

First fame and the Grand tour: 1758–1765
Upon his return to England, Gibbon published his first book, Essai sur l'Étude de la Littérature in 1761, which produced an initial taste of celebrity and distinguished him, in Paris at least, as a man of letters. From 1759 to 1770, Gibbon served on active duty and in reserve with the South Hampshire Militia, his deactivation in December 1762 coinciding with the militia's dispersal at the end of the Seven Years' War. The following year, he returned, via Paris, to Lausanne, where he made the acquaintance of a "prudent worthy young man" William Guise. On 18 April 1764, he and Guise set off for Italy, crossed the Alps, and after spending the summer in Florence arrived in Rome, via Lucca, Pisa, Livorno and Siena, in early October. In his autobiography, Gibbon vividly records his rapture when he finally neared "the great object of [my] pilgrimage":

"...at the distance of twenty-five years I can neither forget nor express the strong emotions which agitated my mind as I first approached and entered the eternal City. After a sleepless night, I trod, with a lofty step the ruins of the Forum; each memorable spot where Romulus stood, or Tully spoke, or Caesar fell, was at once present to my eye; and several days of intoxication were lost or enjoyed before I could descend to a cool and minute investigation."

Here, Gibbon first conceived the idea of composing a history of the city, later extended to the entire empire, a moment he described later as his "Capitoline vision":

"It was at Rome, on the fifteenth of October 1764, as I sat musing amidst the ruins of the Capitol, while the barefooted fryars were singing vespers in the temple of Jupiter, that the idea of writing the decline and fall of the City first started to my mind."

Womersley (Oxford Dictionary of National Biography, p. 12) notes the existence of "good reasons" to doubt the statement's accuracy. Elaborating, Pocock ("Classical History," ¶ #2) refers to it as a likely "creation of memory" or a "literary invention", given that Gibbon, in his autobiography, claimed that his journal dated the reminiscence to 15 October, when in fact the journal gives no date.

Work
In June 1765, Gibbon returned to his father's house, remaining there until the latter's death in 1770. These five years were considered by Gibbon as the worst of his life, but he tried to remain busy by making early attempts at full histories. His first historical narrative, known as the History of Switzerland, representing Gibbon's love for Switzerland, was never finished nor published. Even under the guidance of Deyverdun, his German translator, Gibbon became too self-critical and completely abandoned the project after writing only 60 pages of text.

Soon after abandoning his History of Switzerland, Gibbon made another attempt towards completing a full history. His second work, Memoires Litteraires de la Grande Bretagne, was a two-volume set describing the literary and social conditions of England at the time, such as Lord Lyttelton's history of Henry II and Nathaniel Lardner's The Credibility of the Gospel History. Gibbon's Memoires Litteraires failed to gain any notoriety and was considered a flop by fellow historians and literary scholars.

After he tended to his father's estate—which was in poor condition—enough remained for Gibbon to settle fashionably in London at 7 Bentinck Street free of financial concern. By February 1773, he was writing in earnest, but not without the occasional self-imposed distraction. He took to London society quite easily, joined the better social clubs (including Dr. Johnson's Literary Club), and looked in from time to time on his friend Holroyd in Sussex. He succeeded Oliver Goldsmith at the Royal Academy as 'professor in ancient history', an honorary but prestigious position. In late 1774, he was initiated as a Freemason of the Premier Grand Lodge of England.

He was also, perhaps least productively in that same year, returned to the House of Commons for Liskeard, Cornwall through the intervention of his relative and patron, Edward Eliot. He became the archetypal back-bencher, benignly "mute" and "indifferent," his support of the Whig ministry invariably automatic. Gibbon lost the Liskeard seat in 1780 when Eliot joined the opposition, taking with him "the Electors of Leskeard [who] are commonly of the same opinion as Mr. El[l]iot." (Murray, p. 322.) The following year, owing to the good grace of Prime Minister Lord North, he was again returned to Parliament, this time for Lymington on a by-election.

The History of the Decline and Fall of the Roman Empire: 1776–1788
After several rewrites, with Gibbon "often tempted to throw away the labours of seven years," the first volume of what was to become his life's major achievement, The History of the Decline and Fall of the Roman Empire, was published on 17 February 1776. Through 1777, the reading public eagerly consumed three editions, for which Gibbon was rewarded handsomely: two-thirds of the profits, amounting to approximately £1,000.

Volumes II and III appeared on 1 March 1781, eventually rising "to a level with the previous volume in general esteem." Volume IV was finished in June 1784; the final two were completed during a second Lausanne sojourn (September 1783 to August 1787) where Gibbon reunited with his friend Deyverdun in leisurely comfort. By early 1787, he was "straining for the goal" and with great relief the project was finished in June. Gibbon later wrote:

"It was on the day, or rather the night, of 27 June 1787, between the hours of eleven and twelve, that I wrote the last lines of the last page in a summer-house in my garden...I will not dissemble the first emotions of joy on the recovery of my freedom, and perhaps the establishment of my fame. But my pride was soon humbled, and a sober melancholy was spread over my mind by the idea that I had taken my everlasting leave of an old and agreeable companion, and that, whatsoever might be the future fate of my history, the life of the historian must be short and precarious."

Volumes IV, V, and VI finally reached the press in May 1788, their publication having been delayed since March so it could coincide with a dinner party celebrating Gibbon's 51st birthday (the 8th). Mounting a bandwagon of praise for the later volumes were such contemporary luminaries as Adam Smith, William Robertson, Adam Ferguson, Lord Camden, and Horace Walpole. Adam Smith told Gibbon that "by the universal assent of every man of taste and learning, whom I either know or correspond with, it sets you at the very head of the whole literary tribe at present existing in Europe." In November 1788, he was elected a Fellow of the Royal Society, the main proposer being his good friend Lord Sheffield.

In 1783 Gibbon had been intrigued by the cleverness of Sheffield's 12-year-old eldest daughter, Maria, and he proposed to teach her himself. Over the following years he continued, creating a girl of sixteen who was both well educated, confident and determined to choose her own husband. Gibbon described her as a "mixture of just observation and lively imagery, the strong sense of a man expressed with the easy elegance of a female".

Later life: 1789–1794
The years following Gibbon's completion of The History were filled largely with sorrow and increasing physical discomfort. He had returned to London in late 1787 to oversee the publication process alongside Lord Sheffield. With that accomplished, in 1789 it was back to Lausanne only to learn of and be "deeply affected" by the death of Deyverdun, who had willed Gibbon his home, La Grotte. He resided there with little commotion, took in the local society, received a visit from Sheffield in 1791, and "shared the common abhorrence" of the French Revolution. In 1793, word came of Lady Sheffield's death; Gibbon immediately left Lausanne and set sail to comfort a grieving but composed Sheffield. His health began to fail critically in December, and at the turn of the new year, he was on his last legs.

Among Edward Gibbon's maladies was gout. Gibbon is also believed to have suffered from an extreme case of scrotal swelling, probably a hydrocele testis, a condition that causes the scrotum to swell with fluid in a compartment overlying either testicle. In an age when close-fitting clothes were fashionable, his condition led to a chronic and disfiguring inflammation that left Gibbon a lonely figure. As his condition worsened, he underwent numerous procedures to alleviate the condition, but with no enduring success. In early January, the last of a series of three operations caused an unremitting peritonitis to set in and spread, from which he died.

The "English giant of the Enlightenment" finally succumbed at 12:45 pm, 16 January 1794 at age 56. He was buried in the Sheffield Mausoleum attached to the north transept of the Church of St Mary and St Andrew, Fletching, East Sussex, having died in Fletching while staying with his great friend, Lord Sheffield. Gibbon's estate was valued at approximately £26,000. He left most of his property to cousins. As stipulated in his will, Sheffield oversaw the sale of his library at auction to William Beckford for £950. What happened next suggests that Beckford may have known of Gibbon's moralistic, 'impertinent animadversion' at his expense in the presence of the Duchess of Devonshire at Lausanne. Gibbon's wish that his 6,000-book library would not be locked up 'under the key of a jealous master' was effectively denied by Beckford who retained it in Lausanne until 1801 before inspecting it, then locking it up again until at least as late as 1818 before giving most of the books back to Gibbon's physician Dr Scholl who had helped negotiate the sale in the first place. Beckford's annotated copy of the Decline and Fall turned up in Christie's in 1953, complete with his critique of what he considered the author's 'ludicrous self-complacency ... your frequent distortion of historical Truth to provoke a gibe, or excite a sneer ... your ignorance of oriental languages [etc.]'.

Legacy
A view frequently attributed to Gibbon, that the Roman Empire fell due to its embrace of Christianity, is not widely accepted by scholars today. Gibbon argued that with the empire's new Christian character, large sums of wealth that would have otherwise been used in the secular affairs in promoting the state were transferred to promoting the activities of the Church. However, the pre-Christian empire also spent large financial sums on religious affairs and it is unclear whether or not the change of religion increased the amount of resources the empire spent on religion. Gibbon further argued that new attitudes in Christianity caused many Christians of wealth to renounce their lifestyles and enter a monastic lifestyle, and so stop participating in the support of the empire. However, while many Christians of wealth did become monastics, this paled in comparison to the participants in the imperial bureaucracy. Although Gibbon further pointed out that the importance Christianity placed on peace caused a decline in the number of people serving the military, the decline was so small as to be negligible for the army's effectiveness.

Many scholars argue that Gibbon did not in fact blame Christianity for the empire's fall, rather attributing its decline to the effects of luxury and the consequent erosion of its martial character. Such a view echoes the outlook of the Greek historian Polybius, who similarly explained the decadent Greek world's eclipse by the ascendant Roman Republic in Mediterranean affairs. In this understanding of Gibbon, the process of Rome's decay was well underway before Christian adherents numbered a large proportion of the empire. Hence, although Gibbon might have seen Christianity as hastening Rome's fall, he did not consider it as the root cause.

Gibbon's work has been criticised for its scathing view of the Christian church as laid down in chapters XV and XVI, a situation that resulted in the banning of the book in several countries. Gibbon's was accused of disrespecting, and none too lightly, the character of Christian doctrine, by "treat[ing] the Christian church as a phenomenon of general history, not a special case admitting supernatural explanations and disallowing criticism of its adherents". More specifically, the chapters excoriated the church for "supplanting in an unnecessarily destructive way the great culture that preceded it" and for "the outrage of [practising] religious intolerance and warfare".

Gibbon, in letters to Holroyd and others, expected some type of church-inspired backlash, but the harshness of the ensuing torrents exceeded anything he or his friends had anticipated. Contemporary detractors such as Joseph Priestley and Richard Watson stoked the nascent fire, but the most severe of these attacks was an "acrimonious" piece by the young cleric, Henry Edwards Davis.

Gibbon's apparent antagonism to Christian doctrine spilled over into the Jewish faith, leading to charges of anti-Semitism. For example, he wrote:

"From the reign of Nero to that of Antoninus Pius, the Jews discovered a fierce impatience of the dominion of Rome, which repeatedly broke out in the most furious massacres and insurrections. Humanity is shocked at the recital of the horrid cruelties which they committed in the cities of Egypt, of Cyprus, and of Cyrene, where they dwelt in treacherous friendship with the unsuspecting natives; and we are tempted to applaud the severe retaliation which was exercised by the arms of legions against a race of fanatics, whose dire and credulous superstition seemed to render them the implacable enemies not only of the Roman government, but also of mankind. Womersley, ed., Decline and Fall, vol. 1, ch. XVI, p. 516. see online Gibbon's first footnote here reveals even more about why his detractors reacted so harshly: In Cyrene, [the Jews] massacred 220,000 Greeks; in Cyprus, 240,000; in Egypt, a very great multitude. Many of these unhappy victims were sawed asunder, according to a precedent to which David had given the sanction of his examples. The victorious Jews devoured the flesh, licked up the blood, and twisted the entrails like a girdle around their bodies. see Dion Cassius l. lxviii, p. 1145. As a matter of fact, this is a verbatim citation from Dio Cassius, Historia Romana LXVIII, 32:1–3: The Jewish Uprising : Meanwhile, the Jews in the region of Cyrene had put one Andreas at their head and were destroying both the Romans and the Greeks. They would cook their flesh, make belts for themselves of their entrails, anoint themselves with their blood, and wear their skins for clothing. Many they sawed in two, from the head downwards. Others they would give to wild beasts and force still others to fight as gladiators. In all, consequently, two hundred and twenty thousand perished. In Egypt, also, they performed many similar deeds, and in Cyprus under the leadership of Artemio. There, likewise, two hundred and forty thousand perished. For this reason no Jew may set foot in that land, but even if one of them is driven upon the island by force of the wind, he is put to death. Various persons took part in subduing these Jews, one being Lusius, who was sent by Trajan."

Influence
Gibbon is considered to be a son of the Enlightenment and this is reflected in his famous verdict on the history of the Middle Ages: "I have described the triumph of barbarism and religion." Politically, he rejected the radical egalitarian movements of the time, notably the American and French Revolutions, and dismissed overly rationalistic applications of the rights of man.

Gibbon's work has been praised for its style, his piquant epigrams and its effective irony. Winston Churchill memorably noted in My Early Life, "I set out upon...Gibbon's Decline and Fall of the Roman Empire [and] was immediately dominated both by the story and the style. ...I devoured Gibbon. I rode triumphantly through it from end to end and enjoyed it all." Churchill modelled much of his own literary style on Gibbon's. Like Gibbon, he dedicated himself to producing a "vivid historical narrative, ranging widely over period and place and enriched by analysis and reflection."

Unusually for the 18th century, Gibbon was never content with secondhand accounts when the primary sources were accessible (though most of these were drawn from well-known printed editions). "I have always endeavoured," he says, "to draw from the fountain-head; that my curiosity, as well as a sense of duty, has always urged me to study the originals; and that, if they have sometimes eluded my search, I have carefully marked the secondary evidence, on whose faith a passage or a fact were reduced to depend." In this insistence upon the importance of primary sources, Gibbon is considered by many to be one of the first modern historians:

"In accuracy, thoroughness, lucidity, and comprehensive grasp of a vast subject, the 'History' is unsurpassable. It is the one English history which may be regarded as definitive...Whatever its shortcomings the book is artistically imposing as well as historically unimpeachable as a vast panorama of a great period."

The subject of Gibbon's writing, as well as his ideas and style, have influenced other writers. Besides his influence on Churchill, Gibbon was also a model for Isaac Asimov in his writing of The Foundation Trilogy, which he said involved "a little bit of cribbin' from the works of Edward Gibbon".

Evelyn Waugh admired Gibbon's style, but not his secular viewpoint. In Waugh's 1950 novel Helena, the early Christian author Lactantius worries about the possibility of "'a false historian, with the mind of Cicero or Tacitus and the soul of an animal,' and he nodded towards the gibbon who fretted his golden chain and chattered for fruit."

Monographs by Gibbon

 * Essai sur l’Étude de la Littérature (London: Becket & De Hondt, 1761).
 * Critical Observations on the Sixth Book of [Vergil's] 'The Aeneid' (London: Elmsley, 1770).
 * The History of the Decline and Fall of the Roman Empire (vol. I, 1776; vols. II, III, 1781; vols. IV, V, VI, 1788–1789). all London: Strahan & Cadell.
 * A Vindication of some passages in the fifteenth and sixteenth chapters of the History of the Decline and Fall of the Roman Empire (London: J. Dodsley, 1779).
 * Mémoire Justificatif pour servir de Réponse à l’Exposé, etc. de la Cour de France (London: Harrison & Brooke, 1779).

Other writings by Gibbon

 * "Lettre sur le gouvernement de Berne" [Letter No. IX. Mr. Gibbon to *** on the Government of Berne], in Miscellaneous Works, First (1796) edition, vol. 1 (below). Scholars differ on the date of its composition (Norman, D.M. Low: 1758–59; Pocock: 1763–64).
 * Mémoires Littéraires de la Grande-Bretagne. co-author: Georges Deyverdun (2 vols.: vol. 1, London: Becket & De Hondt, 1767; vol. 2, London: Heydinger, 1768).
 * Miscellaneous Works of Edward Gibbon, Esq., ed. John Lord Sheffield (2 vols., London: Cadell & Davies, 1796; 5 vols., London: J. Murray, 1814; 3 vols., London: J. Murray, 1815). Includes Memoirs of the Life and Writings of Edward Gibbon, Esq..
 * of Edward Gibbon'', ed. John Murray (London: J. Murray, 1896). EG's complete memoirs (six drafts) from the original manuscripts.
 * The Private Letters of Edward Gibbon, 2 vols., ed. Rowland E. Prothero (London: J. Murray, 1896).
 * The works of Edward Gibbon, Volume 3 1906.
 * Gibbon's Journal to 28 January 1763, ed. D.M. Low (London: Chatto and Windus, 1929).
 * Le Journal de Gibbon à Lausanne, ed. Georges A. Bonnard (Lausanne: Librairie de l'Université, 1945).
 * Miscellanea Gibboniana, eds. G.R. de Beer, L. Junod, G.A. Bonnard (Lausanne: Librairie de l'Université, 1952).
 * The Letters of Edward Gibbon, 3 vols., ed. J.E. Norton (London: Cassell & Co., 1956). vol. 1: 1750–1773; vol. 2: 1774–1784; vol. 3: 1784–1794. cited as 'Norton, Letters '.
 * Gibbon's Journey from Geneva to Rome, ed. G.A. Bonnard (London: Thomas Nelson and Sons, 1961). journal.
 * Edward Gibbon: Memoirs of My Life, ed. G.A. Bonnard (New York: Funk & Wagnalls, 1969; 1966). portions of EG's memoirs arranged chronologically, omitting repetition.
 * The English Essays of Edward Gibbon, ed. Patricia Craddock (Oxford: Clarendon Press, 1972); hb: ISBN 0-19-812496-1.

Before 1985

 * Barlow, J. W. (1879). “Gibbon and Julian”. In: Hermathena, Volume 3, 142–159. Dublin: Edward Posonby.
 * Beer, Gavin de. Gibbon and His World. London: Thames and Hudson, 1968. HB: ISBN 0-670-28981-7.
 * Bowersock, G. W., et al. eds. Edward Gibbon and the Decline and Fall of the Roman Empire. Cambridge, MA: Harvard University Press, 1977.
 * Craddock, Patricia B. Young Edward Gibbon: Gentleman of Letters. Baltimore, MD: Johns Hopkins University Press, 1982. HB: ISBN 0-8018-2714-0. Biography.
 * Jordan, David. Gibbon and his Roman Empire. Urbana, IL: University of Illinois Press, 1971.
 * Keynes, Geoffrey, ed. The Library of Edward Gibbon. 2nd ed. Godalming, England: St. Paul's Bibliographies, 1940, repr. 1980.
 * Lewis, Bernard. "Gibbon on Muhammad". Daedalus 105:3 (Summer 1976), 89–101.
 * Low, D. M. Edward Gibbon 1737–1794. London: Chatto and Windus, 1937. Biography.
 * Momigliano, Arnaldo. "Gibbon's Contributions to Historical Method". Historia 2 (1954), 450–463. Reprinted in Momigliano, Studies in Historiography (New York: Harper & Row, 1966; Garland Pubs., 1985), 40–55. PB: ISBN 0-8240-6372-4.
 * Porter, Roger J. "Gibbon's Autobiography: Filling Up the Silent Vacancy". Eighteenth-Century Studies 8:1 (Autumn 1974), 1–26.
 * Stephen, Leslie, "Gibbon's Autobiography" in Studies of a Biographer, Vol. 1 (1898)
 * Swain, J. W. Edward Gibbon the Historian. New York: St. Martin's Press, 1966.
 * White, Jr. Lynn, ed. The Transformation of the Roman World: Gibbon's Problem after Two Centuries. Berkeley: University of California Press, 1966. HB: ISBN 0-520-01334-4.
 * White, Jr. Lynn, ed. The Transformation of the Roman World: Gibbon's Problem after Two Centuries. Berkeley: University of California Press, 1966. HB: ISBN 0-520-01334-4.

Since 1985

 * Berghahn, C.-F., and T. Kinzel, eds., Edward Gibbon im deutschen Sprachraum. Bausteine einer Rezeptionsgeschichte. Heidelberg: Universitätsverlag Winter, 2015.
 * Bowersock, G. W. Gibbon's Historical Imagination. Stanford: Stanford University Press, 1988.
 * Burrow, J. W. Gibbon (Past Masters). Oxford: Oxford University Press, 1985. HB: ISBN 0-19-287553-1. PB: ISBN 0-19-287552-3.
 * Carnochan, W. Bliss. Gibbon's Solitude: The Inward World of the Historian. Stanford: Stanford University Press, 1987. HB: ISBN 0-8047-1363-4.
 * Chaney, Edward, "Reiseerlebnis und 'Traumdeutung' bei Edward Gibbon und William Beckford", Europareisen politisch-sozialer Eliten im 18.Jahrhundert, eds. J. Rees, W. Siebers and H. Tilgner (Berlin 2002), pp. 243–60.
 * Chaney, Edward, "Gibbon, Beckford and the Interpretation of Dreams, Waking Thoughts, and Incidents", The Beckford Society Annual Lectures 2000-2003, ed. Jon Millinton (Beckford Society, 2004).
 * Craddock, Patricia B. Edward Gibbon: a Reference Guide. Boston: G. K. Hall, 1987. PB: ISBN 0-8161-8217-5. A comprehensive listing of secondary literature through 1985. See also her supplement covering the period through 1997.
 * Ghosh, Peter R. "Gibbon Observed". Journal of Roman Studies 81 (1991), 132–156.
 * Ghosh, Peter R. "Gibbon's First Thoughts: Rome, Christianity and the Essai sur l'Étude de la Litterature 1758–61". Journal of Roman Studies 85 (1995), 148–164.
 * Ghosh, Peter R. "The Conception of Gibbon's History", in McKitterick and Quinault, eds. Edward Gibbon and Empire, 271–316.
 * Ghosh, Peter R. "Gibbon's Timeless Verity: Nature and Neo-Classicism in the Late Enlightenment," in Womersley, Burrow, Pocock, eds. Edward Gibbon: bicentenary essays.
 * Ghosh, Peter R. "Gibbon, Edward 1737–1794 British historian of Rome and universal historian," in Kelly Boyd, ed. Encyclopedia of Historians and Historical Writing (Chicago: Fitzroy Dearborn, 1999), 461–463.
 * Kapossy, Béla, Lovis, Béatrice (dir.), Edward Gibbon et Lausanne. Le Pays de Vaud à la rencontre des Lumières européennes. Gollion: Infolio, 2022, 528 p.
 * Levine, Joseph M., "Edward Gibbon and the Quarrel between the Ancients and the Moderns," in Levine, Humanism and History: origins of modern English historiography (Ithaca, NY: Cornell University Press, 1987).
 * Levine, Joseph M. "Truth and Method in Gibbon's Historiography," in Levine, The Autonomy of History: truth and method from Erasmus to Gibbon (Chicago: Chicago University Press, 1999).
 * McKitterick, R., and R. Quinault, eds. Edward Gibbon and Empire. Cambridge: Cambridge University Press, 1997.
 * Norman, Brian. "The Influence of Switzerland on the Life and Writings of Edward Gibbon," in Studies on Voltaire and the Eighteenth Century [SVEC] v.2002:03. Oxford: Voltaire Foundation, 2002.
 * O'Brien, Karen. "English Enlightenment Histories, 1750–c.1815" in.
 * Pocock, J. G. A. Barbarism and Religion, 4 vols.: vol. 1, The Enlightenments of Edward Gibbon, 1737–1764, 1999 [hb: ISBN 0-521-63345-1]; vol. 2, Narratives of Civil Government, 1999 [hb: ISBN 0-521-64002-4]; vol. 3, The First Decline and Fall, 2003 [pb: ISBN 0-521-82445-1]; vol. 4, Barbarians, Savages and Empires, 2005 [pb: ISBN 0-521-72101-6]. all Cambridge Univ. Press.
 * Porter, Roy. Gibbon: Making History. New York: St. Martin's Press, 1989, HB: ISBN 0-312-02728-1.
 * Turnbull, Paul. "'Une marionnette infidele': the Fashioning of Edward Gibbon's Reputation as the English Voltaire," in Womersley, Burrow, Pocock, eds. Edward Gibbon: bicentenary essays.
 * Womersley, David P. The Transformation of The Decline and Fall of the Roman Empire. Cambridge: Cambridge University Press, 1988. HB: ISBN 0-521-35036-0.
 * Womersley, David P., John Burrow, and J. G. A. Pocock, eds. Edward Gibbon: bicentenary essays. Oxford: Voltaire Foundation, 1997. HB: ISBN 0-7294-0552-4.
 * Womersley, David P. Gibbon and the ‘Watchmen of the Holy City’: The Historian and His Reputation, 1776–1815. Oxford: Oxford University Press, 2002. PB: ISBN 0-19-818733-5.

title: Edward Gibbon