title: Recorder (musical instrument)

text: In the 16th century, the structure, repertoire, and performing practice of the recorder is better documented than in prior epochs. The recorder was one of the most important wind instruments of the Renaissance, and many instruments dating to the 16th century survive, including some matched consorts. This period also produced the first extant books describing the recorder, including the treatises of Virdung (1511), Agricola (1529), Ganassi (1535), Cardano (c.1546), Jambe de Fer (1556), and Praetorius (1619). Nonetheless, understanding of the instrument and its practice in this period is still developing.


title: Recorder (musical instrument)

text: In the 16th century, the recorder saw important developments in its structure. As in the recorders of the Middle Ages, the etiology of these changes remains uncertain, development was regional and multiple types of recorder existed simultaneously. Our knowledge is based on documentary sources and surviving instruments.


title: Recorder (musical instrument)

text: Far more recorders survive from the Renaissance than from the Middle Ages. Most of the surviving instruments from the period have a wide, cylindrical bore from the blockline to the uppermost fingerhole, an inverted conical portion down to around the lowest finger hole (the "choke"), then a slight flare to the bell. Externally, they have a curved shape similar to the bore, with a profile like a stretched hourglass. Their sound is warm, rich in harmonics, and somewhat introverted. Surviving consorts of this type, identified by their makers marks, include those marked "HIER S•" or "HIE•S" found in Vienna, Sibiu and Verona; and those marked with variations on a rabbit's footprint, designated "!!" by Adrian Brown, which are dispersed among various museums. The pitch of these recorders is often generally grouped around A = 466 Hz, however little pitch standardization existed in the period. This type of recorder is described by Praetorius in "De Organographia" (1619). A surviving consort by "!!" follows the exact size configuration suggested by Praetorius: stacked fifths up from the basset in F, and down a fifth then a fourth to bass in B and great bass in F. Instruments marked "HIER S•" or "HIE•S" are in stacked fifths from great bass in F to soprano in E. Many of these instruments are pitched around A = 440 Hz or A = 466 Hz, although pitch varied regionally and between consorts.


title: Recorder (musical instrument)

text: The range of this type is normally an octave plus a minor 7th, but as remarked by Praetorius (1619) and demonstrated in the fingering tables of Ganassi's "Fontegara" (1535), experienced players on particular instruments were capable of playing up to a fourth or even a seventh higher (see ). Their range is more suitable for the performance of vocal music, rather than purely instrumental music. This type is the recorder typically referred to as the "normal" Renaissance recorder, however this modern appellation does not fully capture the heterogeneity of instruments of the 16th century.


title: Recorder (musical instrument)

text: Another surviving Renaissance type has a narrow cylindrical bore and cylindrical profile like the medieval exemplars but a choke at the last hole. The earliest surviving recorders of this type were made by the Rafi family, instrument makers active in Lyons in Southern France in the early 16th century. Two recorders marked "C.RAFI" were acquired by the Accademia Filarmonica, Bologna in 1546, where they remain today. A consort of recorders or similar make, marked "P.GRE/C/E," was donated to the Accademia in 1675, expanding the pair marked "C.RAFI". Other recorders by the Rafi family survive in Northern Europe, notably a pair in Brussels. It is possible that Grece worked in the Rafi workshop, or was a member of the Rafi family. The pitch of the Rafi/Grece instruments is around A = 440 Hz. They have a relatively quiet sound with good pitch stability favoring dynamic expression.


title: Recorder (musical instrument)

text: In 1556, French author Philibert Jambe de Fer gave a set of fingerings for hybrid instruments such as the Rafi and Grece instruments that give a range of two octaves. Here, the 15th was now produced, as on most later recorders, as a variant of the 14th instead of as the fourth harmonic of the tonic, as in Ganassi's tables.


title: Recorder (musical instrument)

text: The first two treatises of the 16th century show recorders that differ from the surviving instruments dating to the century: these are Sebastian Virdung's (b. 1465?) (1511), and Martin Agricola's (1486–1556) similar (1529), published in Basel and Saxony respectively.


title: Recorder (musical instrument)

text: , the earliest printed treatise on western musical instruments, is an extract of an earlier, now lost, manuscript treatise by Virdung, a chaplain, singer, and itinerant musician. The printed version was written in a vernacular form of Early New High German, and was aimed at wealthy urban amateur musicians: the title translates, briefly, as "Music, translated into German ... Everything there is to know about [music] – made simple." When a topic become too complex for Virdung to discuss briefly, he refers the reader to his lost larger work, an unhelpful practice for modern readers. While the illustrations have been called "maddeningly inaccurate" and his perspectives quirky, Virdung's treatise gives us an important source on the structure and performing practice of the recorder in northern Europe in the late 15th and early 16th centuries.


title: Recorder (musical instrument)

text: The recorders described by Virdung have cylindrical profiles with flat heads, narrow windows and long ramps, ring-like turnings on the feet, and a slight external flare at the bell (above, far left and middle left). Virdung depicts four recorders together: a or (basset) in F with an anchor shaped key and a perforated fontanelle, two tenors in C and a (alto) in G. According to Virdung, the configurations F–C–C–G or F–C–G–G should be used for four-part music, depending on the range of the bass part. As previously mentioned, the accuracy of these woodcuts cannot be verified as no recorders fitting this description survive. Virdung also provides the first ever fingering chart for a recorder with a range of an octave and a seventh, though he says that the bass had a range of only an octave and sixth. In his fingering chart, he numbers which fingers to lift rather than those to put down and, unlike in later charts, numbers them from bottom (1) to top (8). His only other technical instruction is that the player must blow into the instrument and "learn how to coordinate the articulations ... with the fingers".


title: Recorder (musical instrument)

text: Martin Agricola's ("A German instrumental music, in which is contained how to learn to play ... all kinds of ... instruments"), written in rhyming German verse (ostensibly to improve the understanding and retention of its contents), provides a similar account and copies most of its woodcuts directly from . Agricola also calls the tenor "altus," mistakenly depicting it as a little smaller than the tenor in the woodcut (above, middle right). Like Virdung, Agricola takes it for granted that recorders should be played in four-part consorts. Unlike , which provides a single condensed fingering chart, Agricola provides separate, slightly differing, fingering charts for each instrument, leading some to suppose that Agricola experimented on three different instruments, rather than copying the fingerings from one size to the other two. Agricola adds that graces (), which make the melody , must be learned from a professional (), and that the manner of ornamentation () of the organist is best of all. A substantial 1545 revision of approvingly mentions the use of vibrato () for woodwind instruments, and includes an account of articulation, recommending the syllables ' for semiminims and larger, ' for semiminims and smaller, and the articulation "", which he calls the "flutter-tongue" () for the smallest of note values, found in "passagi (Colorirn)".


title: Recorder (musical instrument)

text: The next treatise comes from Venice: Silvestro Ganassi dal Fontego's (1492–mid-1500s) "Opera Intitulata Fontegara" (1535), which is the first work to focus specifically on the technique of playing the recorder, and perhaps the only historical treatise ever published that approaches a description of a professional or virtuoso playing technique. Ganassi was a musician employed by the Doge and at the Basilica di San Marco at the time of the work's publication, an indication of his high level of accomplishment, and later wrote two works on the playing the viol and the violone, although he does not mention being employed by the Doge after "Fontegara".


title: Recorder (musical instrument)

text: "Fontegara" can be broadly divided into two parts: the first concerns the technique of playing the recorder, the second demonstrated divisions (regole, passagi, ornaments), some of great complexity, which the player may use to ornament a melody or, literally, "divide" it into smaller notes. In all aspects, Ganassi emphasizes the importance of imitating the human voice, declaring that "the aim of the recorder player is to imitate as closely as possible all the capabilities of the human voice", maintaining that the recorder is indeed able to do this. For Ganassi, imitation of the voice has three aspects: "a certain artistic proficiency," which seems to be the ability to perceive the nature of the music, (dexterity or fluency), achieved "by varying the pressure of the breath and shading the tone by means of suitable fingering," and (elegance or grace), achieved by articulation, and by the use of ornaments, the "simplest ingredient" of them being the trill, which varies according to the expression.


title: Recorder (musical instrument)

text: Ganassi gives fingering tables for a range of an octave and a seventh, the standard range also remarked by Praetorius, then tells the reader that he has discovered, through long experimentation, more notes not known to other players due to their lack of perseverance, extending the range to two octaves and a sixth. Ganassi gives fingerings for three recorders with different makers' marks, and advises the reader to experiment with different fingerings, as recorders vary in their bore. The maker's mark of one of the recorders, in the form of a stylized letter "A", has been associated with the Schnitzer family of instrument makers in Germany, leading Hermann Moeck to suppose that Ganassi's recorder might have been Northern European in origin. (see also Note on "Ganassi" recorders)


title: Recorder (musical instrument)

text: Ganassi uses three basic kinds of syllables ', ', and ' and also varies the vowel used with the syllable, suggesting the effect of mouth shape on the sound of the recorder. He gives many combinations of these syllables and vowels, and suggests the choice of the syllables according to their smoothness, ' being least smooth and "" being most so. He does not, however, demonstrate how the syllables should be used to music.


title: Recorder (musical instrument)

text: Most of the treatise consists of tables of diminutions of intervals, small melodies and cadences, categorized by their meter. These several hundred divisions use quintuplets, septuplets, note values from whole notes to 32nd notes in modern notation, and demonstrate immense variety and complexity.


title: Recorder (musical instrument)

text: The frontispiece to "Fontegara" shows three recorder players play together with two singers. Like Agricola and Virdung, Ganassi takes for granted that recorders should be played in groups of four, and come in three sizes: F, C and G. He makes a distinction between solo playing and ensemble playing, noting that what he has said is for solo players, and that when playing with others, it is most important to match them. Unfortunately, Ganassi gives only a few ornamented examples with little context for their use. Nonetheless, Ganassi offers a tantalizing glimpse at a highly developed professional culture and technique of woodwind playing that modern players can scarcely be said to have improved upon.


title: Recorder (musical instrument)

text: Gerolamo Cardano's "De Musica" was written around 1546, but not published until 1663 when it was published along with other works by Cardan, who was an eminent philosopher, mathematician and physician as well as a keen amateur recorder player who learned from a professional teacher, Leo Oglonus, as a child in Milan.


title: Recorder (musical instrument)

text: His account corroborates that of Ganassi, using the same three basic syllables and emphasizing the importance of breath control and ornamentation in recorder playing, but also documents several aspects of recorder technique otherwise undocumented until the 20th century. These include multiple techniques using the partial closing of the bell: to produce a tone or semitone below the tonic, and to change semitones into dieses (half semitones), which he says can also be produced by "repercussively bending back the tongue". He also adds that the position of the tongue, either extended or turned up towards the palate, can be used to improve, vary, and color notes. He is the first to differentiate between the amount of the breath (full, shallow, or moderate) and the force (relaxed or slow, intense, and the median between them) as well as the different amount of air required for each instrument, and describes a trill or vibrato called a in which "a tremulous quality in the breath" is combined with a trilling of the fingers to vary the interval from anything between a major third and a diesis. He is also the first writer to mention the recorder in D ("discantus"), which he leaves unnamed.


title: Recorder (musical instrument)

text: Composer and singer Philibert Jambe de Fer ( 1515 1566) was the only French author of the 16th century to write about the recorder, in his "Epitome musical". He complains of the French name for the instrument, ('flute with nine holes') as, in practice, one of the lowermost holes must be plugged, leaving only eight open holes. He prefers or the Italian . His fingering chart is notable for two reasons, first for describing fingerings with the 15th produced as a variant on the 14th, and for using the third finger of the lower hand as a buttress finger, although only for three notes in the lower octave. (See also Renaissance structure.)


title: Recorder (musical instrument)

text: Aurelio Virgiliano's "" ( 1600) presents ricercars intended for or playable on the recorder, a description of other musical instruments, and a fingering chart for a recorder in G similar to Jambe de Fer's.


title: Recorder (musical instrument)

text: The "Syntagma musicum" (1614–20) of Michael Praetorius (1571–1621) in three volumes (a fourth was intended but never finished) is an encyclopedic survey of music and musical instruments. Volume II, "De Organographia" (1619) is of particular interest for its description of no fewer than eight sizes of recorder ( or "exilent" in G, "discant" in C or D, "alt" in G, "tenor" in C, "basset" in F, "bass" in B, and in F) as well as the four-holed .


title: Recorder (musical instrument)

text: Praetorius was the first author to explain that recorders can confuse the ear into believing that they sound an octave lower than pitch, which phenomenon has more recently been explained in relation to the recorder's lack of high harmonics. He also shows the different "registers" of consort possible, 2′ (discant, alt, and tenor), 4′ (alt, tenor, and basset), and 8′ (tenor, basset, and bass) (see also Nomenclature). Additionally, he proposed cutting the recorder between the beak and the first finger hole to allow for a kind of tuning slide to raise or lower its pitch, similar to the Baroque practice of adjusting a recorder's pitch by "pulling out" the top joint of the recorder.


title: Recorder (musical instrument)

text: The recorders described in Praetorius are of the "stretched hourglass" profile (see above, far right). He gives fingerings like those of Ganassi, and remarks that they normally have a range of an octave and a sixth, although exceptional players could extend that range by a fourth.


title: Recorder (musical instrument)

text: Some paintings from the 14th and 15th centuries depict musicians playing what appear to be two end-blown flutes simultaneously. In some cases, the two flutes are evidently disjoint, separate flutes of similar make, played angled away from each other, one pipe in each hand. In others, flutes of the same length have differing hand positions. In a final case, the pipes are parallel, in contact with each other, and differ in length. While the iconographic criteria for a recorder are typically a clearly recognizable labium and a double handed vertical playing technique, such criteria are not prescriptive, and it is uncertain whether any of these depictions should be considered a single instrument, or constitute a kind of recorder. The identification of the instrument depicted is further complicated by the symbolism of the aulos, a double piped instrument associated with the satyr Marsyas of Greek mythology.


title: Recorder (musical instrument)

text: An instrument consisting of two attached, parallel, end-blown flutes of differing length, dating to the 15th or 16th century, was found in poor condition near All Souls College in Oxford. The instrument has four holes finger-holes and a thumb hole for each hand. The pipes have an inverted conical "choke" bore (see Renaissance structure). Bob Marvin has estimated that the pipes played a fifth apart, at approximately C and G. The instrument is sui generis. Although the instrument's pipes have thumb holes, the lack of organological precedent makes classification of the instrument difficult. Marvin has used the terms "double recorder" and the categorization-agnostic (double flute) to describe the Oxford instrument.


title: Recorder (musical instrument)

text: Marvin has designed a based on the Oxford instrument, scaled to play at F and C. Italian recorder maker Francesco Livirghi has designed a double recorder or with connected, angled pipes of the same length but played with different hand positions, based on iconographic sources. Its pipes play at F and B. Both instruments use fingerings of the makers' design.


title: Recorder (musical instrument)

text: In the 1970s, when recorder makers began to make the first models of recorders from the 16th and 17th centuries, such models were not always representative of the playing characteristics of the original instruments. Especially notable is Fred Morgan's much copied "Ganassi" model, based loosely on an instrument in the Vienna Kunsthistorisches museum (inventory number SAM 135), was designed to use the fingerings for the highest notes in Ganassi's tables in Fontegara. As Morgan knew, these notes were not in standard use; indeed Ganassi uses them in only a few of the hundreds of diminutions contained in Fontegara. Historically, such recorders did not exist as a distinct type, and the fingerings given by Ganassi were those of a skilled player particularly familiar with his instruments. When modern music is written for 'Ganassi recorders' it means this type of recorder.


title: Recorder (musical instrument)

text: Recorders were probably first used to play vocal music, later adding purely instrumental forms such as dance music to their repertoire. Much of the vocal music of the 15th, 16th and 17th centuries can be played on recorder consorts, and as illustrated in treatises from Virdung to Praetorius, the choice appropriate instruments and transpositions to play vocal music was common practice in the Renaissance. Additionally, some collections such as those of Pierre Attaingnant and Anthony Holborne, indicate that their instrumental music was suitable for recorder consorts. This section first discusses repertoire marked for the recorder, then briefly, other repertoire played on recorder.


title: Recorder (musical instrument)

text: In 1505 Giovanni Alvise, a Venetian wind player, offered Francesco Gonzaga of Mantua a motet for eight recorders, however the work has not survived.


title: Recorder (musical instrument)

text: Pierre Attaingnant's ( 1528–1549) (1533) collects 28 (not 27, as in the title) four-part instrumental motets, nine of which he says were suitable for performance on flutes (, German flutes), two on recorders ("," nine-holed flutes, "recorders"), and twelve suitable for both. Of the twelve marked for both, seven use "chiavi naturali", or low-clefs typically used for recorders, while the others use the clefs used in the motets marked for flutes. Hence, the seven notated in could be considered more appropriate for recorders. is the first published music marked for a recorder consort. Earlier is a part for Jacobus Barbireau's song "", apparently for recorder, accompanying the recorder fingering chart in "..." (Antwerp, 1529), a partial French translation of Virdung's .


title: Recorder (musical instrument)

text: Jacques Moderne's published in the 1530s, depicts a four-part recorder consort such as those described in Virdung, Agricola, Ganassi and others, however the dances are not marked for recorders. His (1550) contains ricercares and dances for performance on ", & ".


title: Recorder (musical instrument)

text: In 1539–40, Henry VIII of England, also a keen amateur player (see Cultural significance), imported five brothers of the Bassano family from Venice to form a consort, expanded to six members in 1550, forming a group that maintained an exceptional focus on the recorder until at least 1630 when the recorder consort was combined with the other wind groups. Most wind bands consisted of players playing sackbutts, shawms, and other loud instruments doubling on recorder. Some music probably intended for this group survives, including dance music by Augustine and Geronimo Bassano from the third quarter of the 16th century, and the more elaborate fantasias of Jeronimo Bassano ( 1580), four in five parts and one in six parts. Additionally, the Fitzwilliam wind manuscript ("GB-Cfm" 734) contains wordless motets, madrigals and dance pieces, including some by the Bassano family, probably intended for a recorder consort in six parts.


title: Recorder (musical instrument)

text: The English members of the Bassano family, having originated in Venice, were also probably familiar with the vocal style, advanced technique, and complex improvised ornamentation described in Ganassi's "Fontegara", and they were probably among the recorder players whom Ganassi reports having worked and studied with: when they were brought to England, they were regarded as some of the best wind players in Venice. While most of the music attributed to the consort uses only a range of a thirteenth, it is possible that the Bassano's were familiar with Ganassi's extended range.


title: Recorder (musical instrument)

text: Recorders were also played with other instruments, especially in England, where it was called a mixed consort or "broken consort".


title: Recorder (musical instrument)

text: Other 16th-century composers whose instrumental music can be played well on recorder consorts include:


title: Recorder (musical instrument)

text: The recorder achieved great popularity in the 16th century, and is one of the most common instruments of the Renaissance. From the 15th century onwards, paintings show upper-class men and women playing recorder, and Virdung's didactic treatise (1511), the first of its kind, was aimed at the amateur (see also Documentary evidence). Famously, Henry VIII of England was an avid player of the recorder, and at his death in 1547 an inventory of his possessions included 76 recorders in consorts of various sizes and materials. Some Italian paintings from the 16th-century show aristocracy of both sexes playing the recorder, however many gentlemen found it unbecoming to play because it uses the mouth, preferring the lute and later the viol.


title: Recorder (musical instrument)

text: At the turn of the 17th century, playwright William Shakespeare famously referenced the recorder in his most substantial play, "The Tragedy of Hamlet, Prince of Denmark," creating an extended metaphor between manipulation and playing a musical instrument. Poet John Milton also referenced the recorder in his most famous work, the epic poem Paradise Lost published in 1667, in which the recently fallen angels in Hell "move / in perfect phalanx to the Dorian mood / of flutes and soft recorders," recalling both the affect of the Dorian mode as the mode of calling to action, and the use of flutes by the Spartans of ancient Greece, although the specification of the recorder is anachronistic in this context.


title: Recorder (musical instrument)

text: Several changes in the construction of recorders took place in the 17th century, resulting in the type of instrument generally referred to as "Baroque" recorders, as opposed to the earlier "Renaissance" recorders. These innovations allowed baroque recorders to possess a tone regarded as "sweeter" than that of the earlier instruments, at the expense of a reduction in volume, particularly in the lowest notes.


title: Recorder (musical instrument)

text: The evolution of the Renaissance recorder into the Baroque instrument is generally attributed to the Hotteterre family, in France. They developed the ideas of a more tapered bore, bringing the finger-holes of the lowermost hand closer together, allowing greater range, and enabling the construction of instruments in several jointed sections. The last innovation allowed more accurate shaping of each section and also offered the player minor tuning adjustments, by slightly pulling out one of the sections to lengthen the instrument.


title: Recorder (musical instrument)

text: The French innovations were taken to London by Pierre Bressan, a set of whose instruments survive in the Grosvenor Museum, Chester, as do other examples in various American, European and Japanese museums and private collections. Bressan's contemporary, Thomas Stanesby, was born in Derbyshire but became an instrument maker in London. He and his son (Thomas Stanesby junior) were the other important British-based recorder-makers of the early 18th century.


title: Recorder (musical instrument)

text: In continental Europe, the Denner family of Nuremberg were the most celebrated makers of this period.


title: Recorder (musical instrument)

text: The baroque recorder produces a most brilliant and projecting sound in the second octave, which is more facile and extended than that of earlier recorders, while the lowest notes in its range are relatively weak. Composers such as Bach, Telemann and Vivaldi exploit this property in their concertos for the instrument.


title: Recorder (musical instrument)

text: Measured from its lowest to its highest playable note, the baroque alto recorder has a range of at most two octaves and a fifth with many instruments having a smaller range. Even the most developed instruments of the period, however, cannot produce the augmented tonic, third and fourth of the third octave. Notably, Georg Philipp Telemann's concerto TWV 51:F1 makes use some of these notes in the third octave, posing significant technical challenges to the player, perhaps requiring the covering of the bell or other unusual techniques.


title: Recorder (musical instrument)

text: During the baroque period, the recorder was traditionally associated with pastoral scenes, miraculous events, funerals, marriages, and amorous scenes. Images of recorders can be found in literature and artwork associated with all of these. Purcell, J. S. Bach, Telemann, and Vivaldi used the recorder to suggest shepherds and imitate birds in their music.


title: Recorder (musical instrument)

text: Although the recorder achieved a greater level of standardization in the Baroque than in previous periods, indeed it is the first period in which there was a "standard" size of recorder, ambiguous nomenclature and uncertain organological evidence have led to controversy regarding which instruments should be used in some "flute" parts from the period.


title: Recorder (musical instrument)

text: The concertino group of Bach's fourth Brandenburg Concerto in G major, BWV 1049, consists of a , and , with ripieno strings. His later harpsichord transcription of this concerto, BWV 1057, lowers the key by a tone, as in all of Bach's harpsichord transcriptions, and is scored for solo harpsichord, two and ripieno strings. The desired instrument for the parts in BWV 1049 has been a matter of perennial musicological and organological debate for two primary reasons: first, the term is not mentioned in dictionaries or tutors of the period; and second, the first part uses F#6, a note which is difficult to produce on a Baroque alto recorder in F4.


title: Recorder (musical instrument)

text: The instrumentation of BWV 1057 is uncontroversial: unambiguously specifies recorders, and both parts have been modified to fit comfortably on altos in F4, avoiding, for example, an unplayable Eb4 in the second that would have resulted from a simple transposition of a tone.


title: Recorder (musical instrument)

text: For the first and last movements of the concerto, two opinions predominate: first, that both recorder parts should be played on alto recorders in F4; and second, that the first part should be played on an alto recorder in G and the second part on an alto in F. Tushaar Power has argued for the alto in G4 on the basis that Bach uses the high F#6, which can be easily played on an alto in G4, but not the low F4, a note not playable on the alto in G4. He corroborates this with other alto recorder parts in Bach's cantatas. Michael Marissen reads the repertoire differently, demonstrating that in other recorder parts, Bach used both the low F4 and F#6, as well as higher notes. Marissen argues that Bach was not as consistent as Power asserts, and that Bach would have almost certainly had access to only altos in F. He corroborates this with examinations of pitch standards and notation in Bach's cantatas, in which the recorder parts are sometimes written as transposing instruments to play with organs that sounded as much as a minor third above written pitch. Marissen also reads Bach's revisions to the recorder parts in BWV 1057 as indicative of his avoidance of F#6 in BWV 1049, a sign that he only used the difficult note when necessary in designing the part for an alto recorder in F4. He posits that Bach avoided F#6 in BWV 1049, at the cost of inferior counterpoint, reinstating them as E6 in BWV 1057.


title: Recorder (musical instrument)

text: In the second movement, breaking of beaming in the parts, markings of "f" and "p," the fermata over the final double bar of the first movement, and the 21 bars of rest at the beginning of the third have led some musicologists to argue that Bach intended the use of "echo flutes" distinct from normal recorders in the second movement in particular. The breaking of beaming could be an indication of changes in register or tonal quality, the rests introduced to allow the players time to change instruments, and the markings of "f" and "p" further indicative of register or sound changes. Marissen has demonstrated that the "f" and "p" markings probably indicated tutti and solo sections rather than loud and soft ones.


title: Recorder (musical instrument)

text: A number of instruments other than normal recorders have been suggested for the . One of the earliest proposed alternatives, by Thurston Dart, was the use of double flageolets, a suggestion since revealed to be founded on unsteady musicological grounds. Dart did, however, bring to light numerous newspaper references to Paisible's performance on an "echo flute" between 1713 and 1718. Another contemporary reference to the "echo flute" is in Etienne Loulié's (Amsterdan, 1696): (The sounds of two echo flutes are different, because one is strong and the other is weak). Loulié is unclear on why one would need two echo flutes to play strongly and weakly, and on why it is that echo flutes differ. Perhaps the echo flute was composed in two halves: one which plays strongly, the other weakly? On this we can only speculate.


title: Recorder (musical instrument)

text: Surviving instruments which are candidates for echo flutes include an instrument in Leipzig which consists of two recorders of different tonal characteristics joined at the head and footjoints by brass flanges. There is also evidence of double recorders tuned in thirds, but these are not candidates for the parts in BWV 1049.


title: Recorder (musical instrument)

text: Vivaldi wrote three concertos for the , possibly for performance by students at the Ospedale della Pietà in Venice, where he taught and composed in the early 18th century. They feature virtuosic solo writing, and along with his concerto RV 441 and trio sonata RV 86 are his most virtuosic recorder works. They each survive a single hastily written manuscript copy, each titled (Concerto for little flute) with the additional note (The instruments transpose by a fourth) in RV 443 and (The instruments lower by a fourth) in RV 445. The three concertos RV 443, 444, and 445 are notated in C major, C major and A minor respectively. Also of note is the occasional use of notes outside the normal two octave compass of the recorder: the range of the solo sections is two octaves from notated F4 to notated F6, however there is a single notated C4 in the first movement of RV 444, a notated E4 in a tutti section in the first movement of RV 443 and low E4 in multiple tutti sections of RV 445.


title: Recorder (musical instrument)

text: A number of possible have been proposed as the instrument intended for the performance of these concertos. The first suggestion was the use of the one keyed piccolo, or another small transverse flute, however such instruments had fallen out of use in Venice by the generally accepted time of composition of these concertos in the 1720s, and this opinion is no longer considered well supported. Another suggestion, first proposed by Peter Thalheimer, is the "French" flageolet (see Flageolets below) in G5, which was notated in D4, appearing a fourth lower, possibly explaining the note in the margins of RV 443 and RV 445 () and supported by Bismantova (1677 rev. 1694) and Bonanni (1722) which equate to the flageolet. However this suggestion has been opposed by the presence of notated F and F which are not within the typical compass of the flageolet, although they may be produced through the covering of the bell, sometimes combined with underblowing, as attested by theorists as early as Cardano (c. 1546) and as late as Bellay (c. 1800).


title: Recorder (musical instrument)

text: Two instruments are conventionally accepted today for the performance of these concertos, the sopranino recorder, notated like an alto but sounding an octave higher, and the soprano recorder, following the instruction to transpose the parts down by a fourth. Winfried Michel was first to argue in favor of the soprano recorder in 1983, when he proposed to take Vivaldi at his word and transpose the string parts down a fourth and play the part on a soprano recorder in C5 (also "fifth-flute") using the English practice of notating such flutes as transposing instruments using the fingerings of an alto recorder. Michel notes that this transposition allows for the use of the violins' and viola's lowest strings (in sections where they provide the accompaniment without bass) and the lowest two notes of the 'cello. He attributes the presence of notes not in the recorder's normal compass to Vivaldi's haste, noting that these notes do not appear in the solo sections. He has edited editions of RV 443 and RV 445 for soprano recorder in G major and E minor respectively. Federico Maria Sardelli concurs with Michel in supposing that the margin note was intended to allow the performance of the concertos on the soprano recorder on a specific occasion, however concludes that they were probably written for the sopranino recorder in F5, noting that small transverse flutes had fallen out of use in Italy by Vivaldi's time, the paucity of flageolets in Italy, the range of the parts, and uses of the in vocal arias.


title: Recorder (musical instrument)

text: The recorder was little used in art music of the Classical and Romantic periods. Researchers have long debated why this change occurred, and to what extent the recorder remained use in the late 18th century, and later the 19th century. A significant question in this debate is which, if any, duct flutes of this period are recorders or successors to recorders.


title: Recorder (musical instrument)

text: The recorder work of the latter half of the 18th century most known today is probably a trio sonata by C. P. E. Bach, Wq.163, composed in 1755an arrangement of a trio sonata for two violins and continuo, scored for the unusual ensemble of viola, bass recorder and continuo. This work is also notable for being perhaps the only significant surviving historical solo work for bass recorder. Also of note are the works of Johann Christoph Schultze ( 1733–1813), who wrote two concertos for the instrument, one in G major and another in B major, written around 1740. The last occurrences of the recorder in art music are apparently by Carl Maria von Weber in "Peter Schmoll und seine Nachbarn" (1801) and (1806). Hector Berlioz may have intended "" from "L'enfance du Christ" (1853) for the instrument. Donizetti owned three recorders.


title: Recorder (musical instrument)

text: Many reasons supporting the conventional view that the recorder declined have been proposed. The first significant explanation for the recorder's decline was proposed by Waitzman (1967), who proposed six reasons:


title: Recorder (musical instrument)

text: In the Baroque, the majority of professional recorder players were primarily oboists or string players. For this reason, the number of professional exponents of the recorder was smaller than that of other woodwinds.


title: Recorder (musical instrument)

text: Others attribute the decline of the recorder in part to the flute innovators of the time, such as Grenser, and Tromlitz, who extended the transverse flute's range and evened out its tonal consistency through the addition of keys, or to the supposedly greater dynamic range and volume of the flute. Similar developments occurring in many other orchestral instruments to make them louder, increase their range, and increase their tonal consistency did not simultaneously occur in the case of the recorder.


title: Recorder (musical instrument)

text: A complementary view recently advanced by Nikolaj Tarasov is that the recorder, rather than totally disappearing, evolved in similar ways to other wind instruments through the addition of keys and other devices, and remained in use throughout the 19th century, with its direct descendant's popularity overlapping with the late 19th and early 20th century recorder revival. Support for this view rests on the organological classification of some 19th century duct flutes as recorders. For more on this question, see "Other duct flutes".


title: Recorder (musical instrument)

text: Duct flutes remained popular even as the recorder waned in the 18th century. As in the instrument's earliest history, questions of the instrument's quiddity are at the forefront of modern debate. The modification and renaming of recorders in the 18th century in order to prolong their use, and the uncertainty of the extent of the recorder's use the late 18th and early 19th centuries have fueled these debates. Some recent researchers contend that some 19th century duct flutes are actually recorders. This article briefly discusses the duct flutes presented as successors to the recorder: the English flageolet and the csakan, which were popular among amateurs in the second half of the 18th century, and the whole of the 19th.


title: Recorder (musical instrument)

text: The word "flageolet" has been used since the 16th century to refer to small duct flutes, and the instrument is sometimes designated using general terms such as and , complicating identification of its earliest form. It was first described by Mersenne in (1636) as having four fingers holes on the front and two thumb holes on the back, with a lowest note C6 and a compass of two octaves. Like the recorder, the upper thumb hole is used as an octaving vent. Flageolets were generally small flutes; however, their lowest note varies. They were initially popular in France, and it is from there that the flageolet first arrived in England in the seventeenth century, becoming a popular amateur instrument, as the recorder later did. Indeed, when the recorder was introduced to England, it was presented as an easy instrument for those who already played the flageolet, and the earliest English recorder tutors are notated in the flageolet tablature of the time, called "dot-way". Notably, the diarist and naval administrator Samuel Pepys (1633–1703) and his wife were both amateur players of the flageolet, and Pepys was later an amateur recorder player.


title: Recorder (musical instrument)

text: Starting in the early 1800s, a number of innovations were introduced to the flageolet, including the addition of keys to extend its range and allow it to more easily play accidentals. They also included solutions to the problem of condensation: most commonly, a sea sponge was placed inside the wind chamber (the conical chamber above the windway) to soak up moisture. More novel solutions such as the insertion of a thin wooden wedge into the windway, the drilling of little holes in the side of the block to drain condensation, and a complex system for draining condensation through a hollowed-out block were also developed. Around 1800 in England, the recorder ("English flute," see Name) came to be called an "English flageolet," appropriating the name of the more fashionable instrument. From at least this time to the present, the flageolet in its first form has been called the French flageolet to differentiate it from the so-called English flageolet.


title: Recorder (musical instrument)

text: From around 1803, when the London instrument maker William Bainbridge obtained a number of patents for improvements to the English flageolet, instruments were often referred as "improved" or "patent" flageolets with little reference to how they actually differed from their predecessors. In this period, the instrument had six finger holes and single thumb hole, and had as many as six keys. Tarasov reports that the English flageolets of the late 18th century had six finger holes and no thumb hole, and later regained the thumb hole and seventh finger hole (see above, right). The English flageolet never reached the level of popularity that the "French" flageolet enjoyed in the 19th century, possibly because the latter instrument was louder. Both remained popular until the beginning of the 20th century.


title: Recorder (musical instrument)

text: A significant amount of music was written for the flageolet in the 19th century, such as the etudes of Narcisse Bousquet, although much of it was directed at amateurs.


title: Recorder (musical instrument)

text: English flageolets that may qualify as recorders are of two types: those early instruments, called "English flageolets," which were actually recorders, and 19th century instruments with seven finger holes and a thumb hole. These instruments are not typically regarded as recorders; however, Tarasov has argued for their inclusion in the family.


title: Recorder (musical instrument)

text: The csakan (from Hung. 'pickaxe'), also known by the recorder's old French name , was a duct flute in the shape of a walking stick or oboe popular in Vienna from about 1800 to the 1840s. The csakan was played using the fingerings of a recorder in C, and was typically pitched in A or G and played as a transposing instrument. The first documented appearance of the csakan was at a concert in Budapest on 18 February 1807 in a performance by its billed inventor, Anton Heberle ( 1806–16). Tarasov has contested Heberle's status as the inventor of the instrument, and has argued that the csakan grew out of a Hungarian war hammer of the same name, which was converted into a recorder, perhaps for playing military music. Around 1800, it was highly fashionable for make walking sticks with additional functions (e.g., umbrellas, swords, flutes, oboes, clarinets, horns) although the csakan was the most popular of these, and the only one that became a musical instrument in its own right.


title: Recorder (musical instrument)

text: The earliest instruments were shaped like a walking stick with a mouthpiece in the handle and had no keys, although they could eventually have up to thirteen keys, along with a tuning slide and a device for narrowing the thumb hole. In the 1820s a csakan "in the pleasing shape of an oboe" was introduced in a "simple" form with a single key and a "complex" form with up to twelve keys like those found on contemporaneous flutes. Well-known makers of the csakan included Johann Ziegler and Stephan Koch in Vienna, and Franz Schöllnast in Pressburg. According to accounts left by Schöllnast, the csakan was primarily an amateur instrument, purchased by those who wanted something simple and inexpensive, however there were also accomplished professionals, such as Viennese court oboist Ernst Krähmer (1795–1837) who toured as far afield as Russia, playing the csakan with acclaimed virtuosity.


title: Recorder (musical instrument)

text: Around 400 works for the csakan were published in the first half of the 19th century, mainly for csakan solo, csakan duet or csakan with guitar or piano. The csakan's repertoire has not yet been fully explored. Notable composers for the instrument include Heberle and Krähmer, and Tarasov notes that piano works by Beethoven were arranged for csakan and guitar (Beethoven is reported to have owned a walking-stick csakan). Modern recorder makers such as Bernhard Mollenhauer and Martin Wenner have made csakan copies.


title: Recorder (musical instrument)

text: Similarities in fingering and design make the csakan at least a close relative of the recorder. Accounts of Krähmer's playing, which report his "diminishing and swelling the notes, up to an almost unbelievable loudness" imply a developed technique using shading and alternate fingerings, far beyond a purely amateur culture of house music. Additionally, Tarasov reports that some recorders by Baroque makers were modified, around 1800, through the addition of keys, including a J. C. Denner (1655–1707) basset recorder in Budapest and an alto by Nikolaus Staub (1664–1734) with added G keys, like the D key on a baroque two-key flute. Another modification is the narrowing of the thumb hole, by way of an ivory plug on the J. C. Denner basset and an alto by Benedikt Gahn (1674–1711), to allow it to serve purely as an octaving vent, as found on many flageolets and csakans. These changes may be archetypal to those found on csakans and flageolets, and constitute an inchoate justification for the continuous development of the Baroque recorder into its 19th-century relatives.


title: Recorder (musical instrument)

text: The concept of a recorder "revival" must be considered in the context of the decline of the recorder in the 18th and 19th centuries. The craft of recorder making was continued in some form by a number of families, such as the produced by the Oeggle family, which traces its lineage to the Walch family of recorder makers the careers of the Schlosser family of Zwota. Heinrich Oskar Schlosser (1875–1947) made instruments sold by the firm of Moeck in Celle and helped to design their Tuju series of recorders. The firm Mollenhauer, currently headed by Bernhard Mollenhauer, can trace its origins to historical instrument makers.


title: Recorder (musical instrument)

text: The recorder, if it did persist through the 19th century, did so in a manner quite unlike the success it enjoyed in previous centuries, or that it would enjoy in the century to come in. Among the earliest ensembles to begin use of recorders in the 20th century was the Bogenhauser Künstlerkapelle (Bogenhausen Artists' Band) which from 1890 to 1939 used antique recorders and other instruments to play music of all ages, including arrangements of classical and romantic music. Nonetheless, the recorder was considered primarily an instrument of historical interest.


title: Recorder (musical instrument)

text: The eventual success of the recorder in the modern era is often attributed to Arnold Dolmetsch. While he was responsible for broadening interest in the United Kingdom beyond the small group of early music specialists, Dolmetsch was not solely responsible for the recorder's broader revival. On the continent his efforts were preceded by those of musicians at the Brussels Conservatoire (where Dolmetsch received his training), and by the German Bogenhauser Künstlerkapelle. Also in Germany, the work of Willibald Gurlitt, Werner Danckerts and Gustav Scheck proceeded quite independently of the Dolmetsches.


title: Recorder (musical instrument)

text: Carl Dolmetsch, the son of Arnold Dolmetsch, became one of the first virtuoso recorder players in the 1920s; but more importantly he began to commission recorder works from leading composers of his day, especially for performance at the Haslemere festival which his father ran. Initially as a result of this, and later as a result of the development of a Dutch school of recorder playing led by Kees Otten, the recorder was introduced to serious musicians as a virtuoso solo instrument both in Britain and in northern Europe.


title: Recorder (musical instrument)

text: Among the influential virtuosos who figure in the revival of the recorder as a serious concert instrument in the latter part of the 20th century are Ferdinand Conrad, Kees Otten, Frans Brüggen, Roger Cotte, Hans-Martin Linde, Bernard Krainis, and David Munrow. Brüggen recorded most of the landmarks of the historical repertoire and commissioned a substantial number of new works for the recorder. Munrow's 1975 double album "The Art of the Recorder" remains as an important anthology of recorder music through the ages.


title: Recorder (musical instrument)

text: Among late 20th-century and early 21st-century recorder ensembles, the trio Sour Cream (led by Frans Brüggen), Flautando Köln, the Flanders Recorder Quartet, Amsterdam Loeki Stardust Quartet and Quartet New Generation have programmed remarkable mixtures of historical and contemporary repertoire. Soloists such as Piers Adams, Dan Laurin and Dorothee Oberlinger, Michala Petri, Maurice Steger.


title: Recorder (musical instrument)

text: In the 2012 Charlotte Barbour-Condini became the first recorder player to reach the final of the biennial BBC Young Musician of the Year competition. Recorder player Sophie Westbrooke was a finalist in the 2014 competition.


title: Recorder (musical instrument)

text: The first recorders to be played in the modern period were antique instruments from previous periods. Anecdotally, Arnold Dolmetsch was motivated to make his own recorders after losing a bag containing his antique instruments. Recorders made in the early 20th century were imitative of baroque models in their exterior form, but differed significantly in their structure. Dolmetsch introduced English fingering, the now standard fingering for "baroque" model instruments, and standardized the doubled 6th and 7th holes found on a handful of antique instruments by the English makers Stanesby and Bressan. Dolmetsch instruments generally had a large rectangular windway, unlike the curved windways of all historical instruments, and played at modern pitch.


title: Recorder (musical instrument)

text: Nearly twice as many pieces have been written for the recorder since its modern revival as were written in all previous epochs. Many of these were composed by avant-garde composers of the latter half of the 20th century who used the recorder for the variety of extended techniques which are possible using its open holes and its sensitivity to articulation.


title: Recorder (musical instrument)

text: Modern composers of great stature have written for the recorder, including Paul Hindemith, Luciano Berio, Jürg Baur, Markus Zahnhausen, Josef Tal, John Tavener, Michael Tippett, Benjamin Britten, Leonard Bernstein, Gordon Jacob, Malcolm Arnold, Steven Stucky, Sean Hickey, and Edmund Rubbra.


title: Recorder (musical instrument)

text: Owing to its ubiquity as a teaching instrument and the relative ease of sound production, the recorder has occasionally been used in popular music by groups such as The Beatles; the Rolling Stones (see, for example, "Ruby Tuesday"); Yes, for example, in the song "I've Seen All Good People"; Jefferson Airplane with Grace Slick on "Surrealistic Pillow"; Led Zeppelin ("Stairway to Heaven"); Jimi Hendrix; Siouxsie and the Banshees; Judy Dyble of Fairport Convention; Dido (e.g. "Grafton Street" on "Safe Trip Home"); and Mannheim Steamroller; Ian Anderson (Jethro Tull)


title: Recorder (musical instrument)

text: The trade of recorder making was traditionally transmitted via apprenticeship. Notable historical makers include the Rafi, Schnitzer and Bassano families in the renaissance; Stanesby (Jr. and Sr.), J.C. and J. Denner, Hotteterre, Bressan, Haka, Heitz, Rippert, Rottenburgh, Steenbergen and Terton. Most of these makers also built other wind instruments such as oboes and transverse flutes. Notably, Jacob Denner is credited with the development of the clarinet from the chalumeau.


title: Recorder (musical instrument)

text: Recorder making declined with the instrument's wane in the late 18th century, essentially severing the craft's transmission to the modern age. With few exceptions, the duct flutes manufactured in the 19th and late 18th centuries were intended for amateur or educational use, and were not constructed to the high standard of earlier epochs.


title: Recorder (musical instrument)

text: Arnold Dolmetsch, the first to achieve commercial production in the 20th century, began to build recorders in 1919. While these early recorders played at a low pitch like that of the available originals, he did not strive for exactitude in reproduction, and by the 1930s the Dolmetsch family firm, then under the direction of Arnold's son Carl Dolmetsch, was mass-producing recorders at modern pitch with wide, straight windways, and began to produce bakelite recorders shortly after the Second World War. Nonetheless, the Dolmetsch models were innovative for their time and proved influential, particularly in standardizing the English fingering system now standard for modern baroque-style instruments and doubled 6th and 7th holes, which are rare on antique instruments.


title: Recorder (musical instrument)

text: In Germany, Peter Harlan began to manufacture recorders in the 1920s, primarily for educational use in the youth movement. Following Harlan's success, numerous makers such as Adler and Mollenhauer began commercial production of recorders, fueling an explosion in the instrument's popularity in Germany. These recorders shared little in common with antiques, with large straight windways, anachronistically pitched consorts, modified fingering systems and other innovations.


title: Recorder (musical instrument)

text: In the latter half of the 20th century, historically informed performance practice was on the rise and recorder makers increasingly sought to imitate the sound and character of antiques. The German-American maker Friedrich von Huene was among the first to research recorders held in European collections and produce instruments intended to reproduce the qualities of the antiques. Von Huene and his Australian colleague Frederick Morgan sought to connect the tradition of the historical wind-makers to the modern day with the understanding that doing so creates the best instruments, and those most suited to ancient music.


title: Recorder (musical instrument)

text: Virtually all recorders manufactured today claim ascendancy to an antique model and most makers active today can trace their trade directly to one of these pioneering makers.


title: Recorder (musical instrument)

text: Today, makers maintaining individual workshops include Ammann, Blezinger, Bolton, Boudreau, Breukink, Brown, Coomber, Cranmore, de Paolis, Ehlert, Meyer, Musch, Netsch, Prescott, Rohmer, Takeyama, von Huene, and Wenner. French maker Philippe Bolton created an electroacoustic recorder and is among the last to offer mounted bell-keys and double bell-keys for both tenor and alto recorders. Those bell-keys extend easily the range of the instrument to more than three octaves. Invented by Carl Dolmetsch in 1957, he first used the bell-key system publicly in 1958.


title: Recorder (musical instrument)

text: In the mid-20th century, German composer and music educator Carl Orff popularized the recorder for use in schools as part of Orff-Schulwerk programs in German schools. Orff's five-volume opus of educational music "Music for Children" contains many pieces for recorders, usually scored for other instruments as well.


title: Recorder (musical instrument)

text: Manufacturers have made recorders out of bakelite and other more modern plastics; they are thus easy to produce, hence inexpensive. Because of this, recorders are popular in schools, as they are one of the cheapest instruments to buy in bulk. They are also relatively easy to play at a basic level because sound production needs only breath, and pitch is primarily determined by fingering (though excessive breath pressure will tend to drive the pitch sharp).


title: Recorder (musical instrument)

text: The recorder is a very social instrument. Many recorder players participate in large groups or in one-to-a-part chamber groups, and there is a wide variety of music for such groupings including many modern works. Groups of different sized instruments help to compensate for the limited note range of the individual instruments. Four part arrangements with a soprano, alto, tenor and bass part played on the corresponding recorders are common, although more complex arrangements with multiple parts for each instrument and parts for lower and higher instruments may also be regularly encountered.


title: Non-Euclidean geometry

text: In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.


title: Non-Euclidean geometry

text: The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point "A", which is not on , there is exactly one line through "A" that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through "A" not intersecting , while in elliptic geometry, any line through "A" intersects .


title: Non-Euclidean geometry

text: Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):


title: Non-Euclidean geometry

text: Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.


title: Non-Euclidean geometry

text: The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote "Elements". In the "Elements", Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results (propositions) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the "parallel postulate", which in Euclid's original formulation is:


title: Non-Euclidean geometry

text: If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.


title: Non-Euclidean geometry

text: Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than Euclid's other postulates:


title: Non-Euclidean geometry

text: For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham (Alhazen, 11th century), Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century).


title: Non-Euclidean geometry

text: The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries". These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.


title: Non-Euclidean geometry

text: Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid, which he didn't realize was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the "Elements"." His work was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis.


title: Non-Euclidean geometry

text: Giordano Vitale, in his book "Euclide restituo" (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.


title: Non-Euclidean geometry

text: In a work titled "Euclides ab Omni Naevo Vindicatus" ("Euclid Freed from All Flaws"), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry.


title: Non-Euclidean geometry

text: He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no "logical" contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.


title: Non-Euclidean geometry

text: In 1766 Johann Lambert wrote, but did not publish, "Theorie der Parallellinien" in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a "Lambert quadrilateral", a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.


title: Non-Euclidean geometry

text: At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.


title: Non-Euclidean geometry

text: The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.


title: Non-Euclidean geometry

text: Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.


title: Non-Euclidean geometry

text: Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832 the Hungarian mathematician János Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter "k". Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.


title: Non-Euclidean geometry

text: Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.


title: Non-Euclidean geometry

text: He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines.


title: Non-Euclidean geometry

text: By formulating the geometry in terms of a curvature tensor, Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature.


title: Non-Euclidean geometry

text: It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call "hyperbolic geometry". Several modern authors still consider "non-Euclidean geometry" and "hyperbolic geometry" synonyms.


title: Non-Euclidean geometry

text: Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called the Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articles in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.


title: Non-Euclidean geometry

text: Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term that generally fell out of use). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.


title: Non-Euclidean geometry

text: There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.


title: Non-Euclidean geometry

text: Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry. As the first 28 propositions of Euclid (in "The Elements") do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.


title: Non-Euclidean geometry

text: To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) "must" be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways:


title: Non-Euclidean geometry

text: The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other (called antipodal points) are identified (considered the same). This is also one of the standard models of the real projective plane. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.


title: Non-Euclidean geometry

text: In the elliptic model, for any given line and a point "A", which is not on , all lines through "A" will intersect .


title: Non-Euclidean geometry

text: Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry?". The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.)


title: Non-Euclidean geometry

text: In the hyperbolic model, within a two-dimensional plane, for any given line and a point "A", which is not on , there are infinitely many lines through "A" that do not intersect .


title: Non-Euclidean geometry

text: In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are represented.


title: Non-Euclidean geometry

text: In three dimensions, there are eight models of geometries. There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely anisotropic (i.e. every direction behaves differently).


title: Non-Euclidean geometry

text: Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject of absolute geometry (also called "neutral geometry"). However, the properties that distinguish one geometry from others have historically received the most attention.


title: Non-Euclidean geometry

text: Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:


title: Non-Euclidean geometry

text: Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.


title: Non-Euclidean geometry

text: The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.


title: Non-Euclidean geometry

text: Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.


title: Non-Euclidean geometry

text: The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, "Euclid and his Modern Rivals", written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll, the author of "Alice in Wonderland".


title: Non-Euclidean geometry

text: In analytic geometry a plane is described with Cartesian coordinates : "C" = { ("x,y") : "x", "y" ∈ ℝ }. The points are sometimes identified with complex numbers where ε ∈ { –1, 0, 1}.


title: Non-Euclidean geometry

text: This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number .


title: Non-Euclidean geometry

text: Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions.


title: Non-Euclidean geometry

text: Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry.


title: Non-Euclidean geometry

text: The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number "z" = e can represent a spacetime event one moment into the future of a frame of reference of rapidity "a". Furthermore, multiplication by "z" amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity "a".


title: Non-Euclidean geometry

text: Kinematic study makes use of the dual numbers formula_3 to represent the classical description of motion in absolute time and space:


title: Non-Euclidean geometry

text: Another view of special relativity as a non-Euclidean geometry was advanced by E. B. Wilson and Gilbert Lewis in "Proceedings of the American Academy of Arts and Sciences" in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions.


title: Betteridge's law of headlines

text: Betteridge's law of headlines is an adage that states: "Any headline that ends in a question mark can be answered by the word "no"." It is named after Ian Betteridge, a British technology journalist who wrote about it in 2009, although the principle is much older. It is based on the assumption that if the publishers were confident that the answer was "yes", they would have presented it as an assertion; by presenting it as a question, they are not accountable for whether it is correct or not. The adage does not apply to questions that are more open-ended than strict yes–no questions.


title: Betteridge's law of headlines

text: The maxim has been cited by other names since 1991, when a published compilation of Murphy's law variants called it "Davis's law", a name that also appears online without any explanation of who Davis was. It has also been referred to as the "journalistic principle" and in 2007 was referred to in commentary as "an old truism among journalists".


title: Betteridge's law of headlines

text: Betteridge's name became associated with the concept after he discussed it in a February 2009 article, which examined a previous TechCrunch article that carried the headline "Did Last.fm Just Hand Over User Listening Data to the RIAA?" :


title: Betteridge's law of headlines

text: A similar observation was made by British newspaper editor Andrew Marr in his 2004 book "My Trade", among Marr's suggestions for how a reader should interpret newspaper articles:


title: Betteridge's law of headlines

text: A 2016 study of a sample of academic journals (not news publications) that set out to test Betteridge's law and Hinchliffe's rule (see below) found that few titles were posed as questions and of those, few were yes/no questions and they were more often answered "yes" in the body of the article rather than "no". A 2018 study of 2,585 articles in four academic journals in the field of ecology similarly found that very few titles were posed as questions at all, with 1.82 percent being wh-questions and 2.15 percent being yes/no questions.


title: Betteridge's law of headlines

text: Of the yes/no questions, 44 percent were answered "yes", 34 percent "maybe", and only 22 percent were answered "no". In 2015, a study of 26,000 articles from 13 news sites on the World Wide Web, conducted by a data scientist and published on his blog, found that the majority (54 percent) were yes/no questions, which divided into 20 percent "yes" answers, 17 percent "no" answers and 16 percent whose answers he could not determine (all percentages rounded by Linander).


title: Betteridge's law of headlines

text: Phrasing headlines as questions is a tactic employed by newspapers that do not "have the facts required to buttress the nut graph". Roger Simon characterized the practice as justifying "virtually anything, no matter how unlikely", giving "Hillary to Replace Biden on Ticket?" and "Romney to Endorse Gay Marriage Between Corporations?" as hypothetical examples of such a practice. Many question headlines were used, for example, in reporting of Bharatiya Janata Party in-fighting in 2004, because no politicians went on record, to confirm or deny facts, such as "Is Venkaiah Naidu on his way out?" Because this implication is known to readers, guides giving advice to newspaper editors state that so-called "question heads" should be used sparingly.


title: Betteridge's law of headlines

text: Freelance writer R. Thomas Berner calls them "gimmickry". Grant Milnor Hyde observed that they give the impression of uncertainty in a newspaper's content. When Linton Andrews worked at the "Daily Mail" after the First World War, one of the rules set by Lord Northcliffe was to avoid question headlines, unless the question itself reflected a national issue.


title: Betteridge's law of headlines

text: Question headlines are not legally sound when it comes to avoiding defamation. The Supreme Court of Oklahoma held in 1913, in its decision in "Spencer v. Minnick", that "A man cannot libel another by the publication of language the meaning and damaging effect of which is clear to all men, and where the identity of the person meant cannot be doubted, and then escape liability through the use of a question mark." The use of question headlines as a form of sensationalism has a long history, including the 9 June 1883, headline in Joseph Pulitzer's "New York World", "Was It Peppermint Mary?"


title: Betteridge's law of headlines

text: The story, about a jewellery store that had tried to prevent its female employees from flirting with people outside the store, only mentioned "Peppermint" Mary at the end of the piece as an employee who might possibly have caused this and did not answer the question.


title: Betteridge's law of headlines

text: The "New York World" also famously used a question headline for hedging when editors were unsure of their facts, when it reported the outcome of the 1916 United States presidential election. When other New York City newspapers ran statement headlines on 8 November 1916 saying "Hughes Is Elected" ("The Evening Sun", final edition the night before), "Hughes Is Elected by Narrow Margin" ("The Sun"), "Hughes Is Elected by Majority of 40" ("The New York Herald"), "Hughes the Next President" ("The Journal of Commerce"), "Hughes Sweeps State" ("New York Tribune") and "Nation Swept by Hughes!" ("New York American"), the "World" ran one with a question headline, "Hughes Elected in Close Contest?"


title: Betteridge's law of headlines

text: This was the result of a last-minute intervention by then "World" journalist Herbert Bayard Swope, who, having received a tip from gambling friends that Charles Evans Hughes might not in fact win, persuaded Charles M. Lincoln, the managing editor of the paper, to reset the headline in between editions, inserting a question mark. Confusingly, below the question headline the "World" still had a picture of Hughes captioned "The President-Elect" but the question headline did indeed turn out to have the answer "no", as President Woodrow Wilson was re-elected, which the "World" finally announced in a headline two days later.


title: Betteridge's law of headlines

text: Advertisers and marketers prefer yes/no question headlines that are answered "yes", as a reader that immediately answers "no" to a question headline on an advertisement is likely to skip over the advertisement entirely. The most famous example of such a question headline in advertising is "Do you make these mistakes in English?", written to advertise Sherwin Cody's English-language course and used from 1919 to 1959, which (with readers answering "yes" they did make the mistakes that the advertisement proceeded to outline) was measured as more successful than non-yes/no-question alternatives.


title: Betteridge's law of headlines

text: Victor Schwab, a partner in the advertising agency that worked for Cody, published an analysis of the aspects of the headline (as ) attempting to look at it scientifically and using ten years' worth of revenue and customer enquiry data for both it and a statement headline that Cody had also used. He noted amongst other things that working in its favour was the question addressing the reader using the second person. A 2013 study into computer-mediated communication came to a similar conclusion, finding that question headlines posted to Twitter and eBay increased click-through rates in comparison to statement headlines and that questions that address or reference the reader have statistically significant higher click-through rates than rhetorical or general questions.


title: Betteridge's law of headlines

text: In the field of particle physics, the concept is known as Hinchliffe's rule, after physicist Ian Hinchliffe, who stated that if a research paper's title is in the form of a yes–no question, the answer to that question will be "no". The adage led into a humorous attempt at a liar paradox by a 1988 paper, written by physicist Boris Kayser under the pseudonym "Boris Peon", which bore the title: "Is Hinchliffe's Rule True?" .


title: Trident

text: The trident is the weapon of Poseidon, or Neptune, the God of the Sea in classical mythology. The trident may occasionally be held by other marine divinities such as Tritons in classical art. Tridents are also depicted in medieval heraldry, sometimes held by a merman-Triton. In Hinduism, it is the weapon of Shiva, known as "trishula" (Sanskrit for "triple-spear").


title: Trident

text: The word "trident" comes from the French word "trident", which in turn comes from the Latin word ' or ': "tri" meaning "three" and "dentes" meaning "teeth", referring specifically to the three prongs, or "teeth", of the weapon.


title: Trident

text: The Greek equivalent is ("tríaina"), from Proto-Greek "trianja", meaning "threefold". The Greek term does not imply three of anything specific, and is vague about the shape, thus the assumption it was originally of "trident" form has been challenged.


title: Trident

text: The Sanskrit name for the trident, "trishula", is a compound of "tri" त्रि for "three" and "śūla" शूल for "thorn", calling the trident's three prongs "thorns" rather than "teeth".


title: Trident

text: The trident is associated with the sea god Poseidon. This divine instrument is said to have been forged by the cyclopes.


title: Trident

text: Poseidon struck a rock with his trident, causing a sea (or a saltwater spring, called the Erechtheis) to appear nearby on the Acropolis in Athens. And according to Roman sources, Neptune struck the earth with the trident to produce the first warhorse.


title: Trident

text: Poseidon, as well as being the god of the sea, was also known as the "Earth Shaker", believed to cause earthquakes; some commentators have extrapolated that the god would have used the trident to cause them, possibly by striking the earth.


title: Trident

text: In the Renaissance artist Gian Bernini's sculpture "Neptune and Triton" (1622–23), Neptune is posed holding a trident turned downwards, and is thought to reenact a scene from "Aeneid" or Ovid's "Metamorphoses" where he is calming the waves to aid Aeneas's ships.


title: Trident

text: In later Greek and Roman art and literature, other sea deities and their attendants have been depicted holding the trident.


title: Trident

text: Poseidon's consort Amphitrite is often identified by some marine attribute other than a trident, which she never carries according to some scholars, though other commentators have disagreed.


title: Trident

text: Turning to the retinue or a train of beings which follow the sea deities (the marine thiasos) the Tritons (mermen) may be seen bearing tridents. Likewise, the Old Man of the Sea ("halios geron") and the god Nereus are seen holding tridents. Tritons, other mermen, and the Nereides can also carry rudders, oars, fish, or dolphins.


title: Trident

text: Oceanus normally should not carry a trident, allowing him to be clearly distinguished from Poseidon. However, there is conflation of the deities in Romano-British iconography, and examples exist where the crab-claw headed Oceanus also bears a trident. Oceanus holding a trident has been found on Romano-British coinage as well.


title: Trident

text: In Hindu legends and stories Shiva, the Hindu god uses a trishula as his principal weapon. The trident is also said to represent three gunas mentioned in Indian Vedic philosophy namely sāttvika, rājasika, and tāmasika. The goddess Kali is sometimes portrayed with a trident as well.


title: Trident

text: In religious Taoism, the trident represents the Taoist Trinity, the Three Pure Ones. In Taoist rituals, a trident bell is used to invite the presence of deities and summon spirits, as the trident signifies the highest authority of Heaven.


title: Trident

text: In heraldry within the UK, the trident is often held by the figure identified as either a Neptune or a triton, or a merman.


title: Trident

text: In Ancient Greece, the trident was employed as a harpoon for spearing large fish, especially tuna fishing.


title: Trident

text: Tridents used in modern spear-fishing usually have barbed tines, which trap the speared fish firmly. In the Southern and Midwestern United States, "gigging" is used for harvesting suckers, bullfrogs, flounder and many species of rough fish.


title: Trident

text: It has been used by farmers as a decorticator to remove leaves, seeds and buds from the stalks of plants such as flax and hemp. A form of trident is used by the gardians in the Camargue of southern France for herding cattle.


title: Trident

text: In Ancient Rome tridents ( or ) were used by a type of gladiator called a "retiarius" or "net fighter". The "retiarius" was traditionally pitted against a "secutor", and cast a net to wrap his adversary and then used the trident to fight him.


title: Trident

text: The trident, known as dangpa, is used as a weapon in the 17th- to 18th-century systems of Korean martial arts.


title: Trident

text: The glyph or sigil of the planet Neptune (♆), which alludes to the trident, is used in astronomy and astrology.


title: Trident

text: A number of structures in the biological world are described as "trident" in appearance. Since at least the late 19th century the trident shape was applied to certain botanical shapes; for example, certain orchid flora were described as having trident-tipped lips in early botanical works. Furthermore, in current botanical literature, certain bracts are stated to have a trident-shape (e.g. Douglas-fir).


title: Likelihood function

text: The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more "likely" than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as formula_1 instead of formula_2, to emphasize that it is to be understood as a function of the parameters formula_3 instead of the random variable formula_4.


title: Likelihood function

text: In maximum likelihood estimation, the arg max of the likelihood function serves as a point estimate for formula_3, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' rule.


title: Likelihood function

text: The likelihood function, parameterized by a (possibly multivariate) parameter formula_3, is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function


title: Likelihood function

text: In other words, when formula_12 is viewed as a function of formula_8 with formula_3 fixed, it is a probability density function, and when viewed as a function of formula_3 with formula_8 fixed, it is a likelihood function. The likelihood function does "not" specify the probability that formula_3 is the truth, given the observed sample formula_18. Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy).


title: Likelihood function

text: Let formula_4 be a discrete random variable with probability mass function formula_20 depending on a parameter formula_3. Then the function


title: Likelihood function

text: considered as a function of formula_3, is the "likelihood function", given the outcome formula_8 of the random variable formula_4. Sometimes the probability of "the value formula_8 of formula_4 for the parameter value formula_3" is written as or . The likelihood is the probability that a particular outcome formula_8 is observed when the true value of the parameter is formula_3, equivalent to the probability mass on formula_8; it is "not" a probability density over the parameter formula_3. The likelihood, formula_33, should not be confused with formula_34, which is the posterior probability of formula_3 given the data formula_8.


title: Likelihood function

text: Consider a simple statistical model of a coin flip: a single parameter formula_37 that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. formula_37 can take on any value within the range 0.0 to 1.0. For a perfectly fair coin, formula_39.


title: Likelihood function

text: Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is i.i.d., then the probability of observing HH is


title: Likelihood function

text: This is not the same as saying that formula_43, a conclusion which could only be reached via Bayes' theorem given knowledge about the marginal probabilities formula_44 and formula_45.


title: Likelihood function

text: Now suppose that the coin is not a fair coin, but instead that formula_46. Then the probability of two heads on two flips is


title: Likelihood function

text: More generally, for each value of formula_37, we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure 1. Note that the integral of formula_50 over [0, 1] is 1/3; likelihoods need not integrate or sum to one over the parameter space.


title: Likelihood function

text: Let formula_4 be a random variable following an absolutely continuous probability distribution with density function formula_52 (a function of formula_8) which depends on a parameter formula_3. Then the function


title: Likelihood function

text: considered as a function of formula_3, is the "likelihood function" (of formula_3, given the outcome formula_58). Again, note that formula_50 is not a probability density or mass function over formula_3, despite being a function of formula_3 given the observation formula_18.


title: Likelihood function

text: The use of the probability density in specifying the likelihood function above is justified as follows. Given an observation formula_63, the likelihood for the interval formula_64, where formula_65 is a constant, is given by formula_66. Observe that


title: Likelihood function

text: and so maximizing the probability density at formula_75 amounts to maximizing the likelihood of the specific observation formula_75.


title: Likelihood function

text: In measure-theoretic probability theory, the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. The likelihood function is this density interpreted as a function of the parameter, rather than the random variable. Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.)


title: Likelihood function

text: The above discussion of the likelihood for discrete random variables uses the counting measure, under which the probability density at any outcome equals the probability of that outcome.


title: Likelihood function

text: The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses formula_77 and a density formula_12, where the sum of all the formula_20's added to the integral of formula_52 is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply


title: Likelihood function

text: where formula_82 is the index of the discrete probability mass corresponding to observation formula_8, because maximizing the probability mass (or probability) at formula_8 amounts to maximizing the likelihood of the specific observation.


title: Likelihood function

text: The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation formula_8, but not with the parameter formula_3.