Apollonius of Perga

Apollonius of Perga ( "Apollonios o Pergeos"; ; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
Apollonius worked on numerous other topics, including astronomy. Most of this work has not survived, where exceptions are typically fragments referenced by other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance. The Apollonius crater on the Moon is named in his honor.
Life.
For such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Greek commentator, Eutocius of Ascalon, on Apollonius’ major work, "Conics", states:
Apollonius, the geometrician, ... came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes ...
Perga at the time was a Hellenized city of Pamphylia in Anatolia. The ruins of the city yet stand. It was a center of Hellenistic culture. Euergetes, "benefactor", identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC. Times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain. The approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the various scholars are only speculative.
Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt. Never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Kingdom of Pergamon to the north, ruled by the Attalid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there. To the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria.
A letter by the Greek mathematician and astronomer Hypsicles was originally part of the supplement taken from Euclid's Book XIV, part of the thirteen books of Euclid's Elements.
The times of Apollonius.
Apollonius lived toward the end of a historical period now termed the Hellenistic Period, characterized by the superposition of Hellenic culture over extensive non-Hellenic regions to various depths, radical in some places, hardly at all in others. The change was initiated by Philip II of Macedon and his son, Alexander the Great, who, subjecting all of Greece in a series of stunning victories, went on to conquer the Persian Empire, which ruled territories from Egypt to Pakistan. Philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Persian empire.
The short autobiography of Apollonius.
The material is located in the surviving false “Prefaces” of the books of his "Conics." These are letters delivered to influential friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that "Conics" was initially requested by a house guest at Alexandria, the geometer, Naucrates, otherwise unknown to history. Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” ("ou diakatharantes" in Greek, "ea non perpurgaremus" in Latin). He intended to verify and emend the books, releasing each one as it was completed.
Hearing of this plan from Apollonius himself on a subsequent visit of the latter to Pergamon, Eudemus had insisted Apollonius send him each book before release. The circumstances imply that at this stage Apollonius was a young geometer seeking the company and advice of established professionals. Pappus states that he was with the students of Euclid at Alexandria. Euclid was long gone. This stay had been, perhaps, the final stage of Apollonius’ education. Eudemus was perhaps a senior figure in his earlier education at Pergamon; in any case, there is reason to believe that he was or became the head of the Library and Research Center (Museum) of Pergamon. Apollonius goes on to state that the first four books were concerned with the development of elements while the last four were concerned with special topics.
There is something of a gap between Prefaces I and II. Apollonius has sent his son, also Apollonius, to deliver II. He speaks with more confidence, suggesting that Eudemus use the book in special study groups, which implies that Eudemus was a senior figure, if not the headmaster, in the research center. Research in such institutions, which followed the model of the Lycaeum of Aristotle at Athens, due to the residency of Alexander the Great and his companions in its northern branch, was part of the educational effort, to which the library and museum were adjunct. There was only one such school in the state. Owned by the king, it was under royal patronage, which was typically jealous, enthusiastic, and participatory. The kings bought, begged, borrowed and stole the precious books whenever and wherever they could. Books were of the highest value, affordable only to wealthy patrons. Collecting them was a royal obligation. Pergamon was known for its parchment industry, whence “parchment” is derived from “Pergamon.”
Apollonius brings to mind Philonides of Laodicea, a geometer whom he introduced to Eudemus in Ephesus. Philonides became Eudemus' student. He lived mainly in Syria during the 1st half of the 2nd century BC. Whether the meeting indicates that Apollonius now lived in Ephesus is unresolved. The intellectual community of the Mediterranean was international in culture. Scholars were mobile in seeking employment. They all communicated via some sort of postal service, public or private. Surviving letters are abundant. They visited each other, read each other's works, made suggestions to each other, recommended students and accumulated a tradition termed by some “the golden age of mathematics.”
Preface III is missing. During the interval Eudemus passed away, says Apollonius in IV, again supporting a view that Eudemus was senior over Apollonius. Prefaces IV–VII are more formal, omitting personal information and concentrating on summarizing the books. They are all addressed to a mysterious Attalus, a choice made “because”, as Apollonius writes to Attalus, “of your earnest desire to possess my works.” By that time a good many people at Pergamum had such a desire. Presumably, this Attalus was someone special, receiving copies of Apollonius’ masterpiece fresh from the author's hand. One strong theory is that Attalus is Attalus II Philadelphus, 220-138 BC, general and defender of his brother's kingdom (Eumenes II), co-regent on the latter's illness in 160 BC, and heir to his throne and his widow in 158 BC. He and his brother were great patrons of the arts, expanding the library into international magnificence. The dates are consonant with those of Philonides, while Apollonius’ motive is consonant with Attalus' book-collecting initiative.
Apollonius sent to Attalus Prefaces V–VII. In Preface VII he describes Book VIII as “an appendix” ... “which I will take care to send you as speedily as possible.” There is no record that it was ever sent or ever completed. It may be missing from history because it was never in history, Apollonius having died before its completion. Pappus of Alexandria, however, provided lemmas for it, so at the very least some edition of it must once have been in circulation.
Documented works of Apollonius.
Apollonius was a prolific geometer, turning out a large number of works. Only one survives, "Conics". It is a dense and extensive reference work on the topic, even by today's standards, serving as a repository of now little known geometric propositions as well as a vehicle for some new ones devised by Apollonius. Its audience was not the general population, which could not read or write. It was always intended for savants of mathematics and their small number of educated readers associated with the state schools and their associated libraries. It always was, in other words, a library reference work. Its basic definitions have become an important mathematical heritage. For the most part its methods and conclusions have been superseded by Analytic Geometry.
Of its eight books, only the first four have a credible claim to descent from the original texts of Apollonius. Books 5-7 have been translated from the Arabic into Latin. The original Greek has been lost. The status of Book VIII is unknown. A first draft existed. Whether the final draft was ever produced is not known. A "reconstruction" of it by Edmond Halley exists in Latin. There is no way to know how much of it, if any, is verisimilar to Apollonius. Halley also reconstructed "De Rationis Sectione" and "De Spatii Sectione". Beyond these works, except for a handful of fragments, documentation that might in any way be interpreted as descending from Apollonius ends.
Many of the lost works are described or mentioned by commentators. In addition are ideas attributed to Apollonius by other authors without documentation. Credible or not, they are hearsay. Some authors identify Apollonius as the author of certain ideas, consequently named after him. Others attempt to express Apollonius in modern notation or phraseology with indeterminate degrees of fidelity.
"Conics".
The Greek text of "Conics" uses the Euclidean arrangement of definitions, figures and their parts; i.e., the “givens,” followed by propositions “to be proved.” Books I-VII present 387 propositions. This type of arrangement can be seen in any modern geometry textbook of the traditional subject matter. As in any course of mathematics, the material is very dense and consideration of it, necessarily slow. Apollonius had a plan for each book, which is partly described in the "Prefaces". The headings, or pointers to the plan, are somewhat in deficit, Apollonius having depended more on the logical flow of the topics.
An intellectual niche is thus created for the commentators of the ages. Each must present Apollonius in the most lucid and relevant way for his own times. They use a variety of methods: annotation, extensive prefatory material, different formats, additional drawings, superficial reorganization by the addition of capita, and so on. There are subtle variations in interpretation. The modern English speaker encounters a lack of material in English due to the preference for New Latin by English scholars. Such intellectual English giants as Edmund Halley and Isaac Newton, the proper descendants of the Hellenistic tradition of mathematics and astronomy, can only be read and interpreted in translation by populations of English speakers unacquainted with the classical languages; that is, most of them.
Presentations written entirely in native English begin in the late 19th century. Of special note is Heath's "Treatise on Conic Sections". His extensive prefatory commentary includes such items as a lexicon of Apollonian geometric terms giving the Greek, the meanings, and usage. Commenting that “the apparently portentious bulk of the treatise has deterred many from attempting to make its acquaintance,” he promises to add headings, changing the organization superficially, and to clarify the text with modern notation. His work thus references two systems of organization, his own and Apollonius’, to which concordances are given in parentheses.
Heath's work is indispensable. He taught throughout the early 20th century, passing away in 1940, but meanwhile another point of view was developing. St. John's College (Annapolis/Santa Fe), which had been a military school since colonial times, preceding the United States Naval Academy at Annapolis, Maryland, to which it is adjacent, in 1936 lost its accreditation and was on the brink of bankruptcy. In desperation the board summoned Stringfellow Barr and Scott Buchanan from the University of Chicago, where they had been developing a new theoretical program for instruction of the Classics. Leaping at the opportunity, in 1937 they instituted the “new program” at St. John's, later dubbed the Great Books program, a fixed curriculum that would teach the works of select key contributors to the culture of western civilization. At St. John's, Apollonius came to be taught as himself, not as some adjunct to analytic geometry.
The “tutor” of Apollonius was R. Catesby Taliaferro, a new PhD in 1937 from the University of Virginia. He tutored until 1942 and then later for one year in 1948, supplying the English translations by himself, translating Ptolemy's "Almagest" and Apollonius’ "Conics". These translations became part of the Encyclopædia Britannica's Great Books of the Western World series. Only Books I-III are included, with an appendix for special topics. Unlike Heath, Taliaferro did not attempt to reorganize Apollonius, even superficially, or to rewrite him. His translation into modern English follows the Greek fairly closely. He does use modern geometric notation to some degree.
Contemporaneously with Taliaferro's work, Ivor Thomas an Oxford don of the World War II era, was taking an intense interest in Greek mathematics. He planned a compendium of selections, which came to fruition during his military service as an officer in the Royal Norfolk Regiment. After the war it found a home in the Loeb Classical Library, where it occupies two volumes, all translated by Thomas, with the Greek on one side of the page and the English on the other, as is customary for the Loeb series. Thomas' work has served as a handbook for the golden age of Greek mathematics. For Apollonius he only includes mainly those portions of Book I that define the sections.
Heath, Taliaferro, and Thomas satisfied the public demand for Apollonius in translation for most of the 20th century. The subject moves on. More recent translations and studies incorporate new information and points of view as well as examine the old.
Book I.
Book I presents 58 propositions. Its most salient content is all the basic definitions concerning cones and conic sections. These definitions are not exactly the same as the modern ones of the same words. Etymologically the modern words derive from the ancient, but the etymon often differs in meaning from its reflex.
A conical surface is generated by a line segment rotated about a bisector point such that the end points trace circles, each in its own plane. A cone, one branch of the double conical surface, is the surface with the point (apex or vertex), the circle (base), and the axis, a line joining vertex and center of base.
A “section” (Latin sectio, Greek tome) is an imaginary “cutting” of a cone by a plane.
The Greek geometers were interested in laying out select figures from their inventory in various applications of engineering and architecture, as the great inventors, such as Archimedes, were accustomed to doing. A demand for conic sections existed then and exists now. The development of mathematical characterization had moved geometry in the direction of Greek geometric algebra, which visually features such algebraic fundamentals as assigning values to line segments as variables. They used a coordinate system intermediate between a grid of measurements and the Cartesian coordinate system. The theories of proportion and application of areas allowed the development of visual equations. (See below under Methods of Apollonius).
The “application of areas” implicitly asks, given an area and a line segment, does this area apply; that is, is it equal to, the square on the segment? If yes, an applicability (parabole) has been established. Apollonius followed Euclid in asking if a rectangle on the abscissa of any point on the section applies to the square of the ordinate. If it does, his word-equation is the equivalent of formula_1 which is one modern form of the equation for a parabola. The rectangle has sides formula_2 and formula_3. It was he who accordingly named the figure, parabola, “application.”
The “no applicability” case is further divided into two possibilities. Given a function, formula_4, such that, in the applicability case, formula_5, in the no applicability case, either formula_6 or formula_7. In the former, formula_8 falls short of formula_9 by a quantity termed the ellipsis, ”deficit.” In the latter, formula_8 overshoots by a quantity termed the hyperbole, "surfeit."
Applicability could be achieved by adding the deficit, formula_11, or subtracting the surfeit, formula_12. The figure compensating for a deficit was named an ellipse; for a surfeit, a hyperbola. The terms of the modern equation depend on the translation and rotation of the figure from the origin, but the general equation for an ellipse,
can be placed in the form
formula_13
where C/B is the d, while an equation for the hyperbola,
becomes
formula_14
where C/B is the s.
Book II.
Book II contains 53 propositions. Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things ... for limits of possibility." His definition of "diameter" is different from the traditional, as he finds it necessary to refer the intended recipient of the letter to his work for a definition. The elements mentioned are those that specify the shape and generation of the figures. Tangents are covered at the end of the book.
Book III.
Book III contains 56 propositions. Apollonius claims original discovery for theorems "of use for the construction of solid loci ... the three-line and four-line locus ..." The locus of a conic section is the section. The three-line locus problem (as stated by Taliafero's appendix to Book III) finds "the locus of points whose distances from three given fixed straight lines ... are such that the square of one of the distances is always in a constant ratio to the rectangle contained by the other two distances." This is the proof of the application of areas resulting in the parabola. The four-line problem results in the ellipse and hyperbola. Analytic geometry derives the same loci from simpler criteria supported by algebra, rather than geometry, for which Descartes was highly praised. He supersedes Apollonius in his methods.
Book IV.
Book IV contains 57 propositions. The first sent to Attalus, rather than to Eudemus, it thus represents his more mature geometric thought. The topic is rather specialized: "the greatest number of points at which sections of a cone can meet one another, or meet a circumference of a circle, ..." Nevertheless, he speaks with enthusiasm, labeling them "of considerable use" in solving problems (Preface 4).
Book V.
Book V, known only through translation from the Arabic, contains 77 propositions, the most of any book. They cover the ellipse (50 propositions), the parabola (22), and the hyperbola (28). These are not explicitly the topic, which in Prefaces I and V Apollonius states to be maximum and minimum lines. These terms are not explained. In contrast to Book I, Book V contains no definitions and no explanation.
The ambiguity has served as a magnet to exegetes of Apollonius, who must interpret without sure knowledge of the meaning of the book's major terms. Until recently Heath's view prevailed: the lines are to be treated as normals to the sections. A normal in this case is the perpendicular to a curve at a tangent point sometimes called the foot. If a section is plotted according to Apollonius’ coordinate system (see below under Methods of Apollonius), with the diameter (translated by Heath as the axis) on the x-axis and the vertex at the origin on the left, the phraseology of the propositions indicates that the minima/maxima are to be found between the section and the axis. Heath is led into his view by consideration of a fixed point p on the section serving both as tangent point and as one end of the line. The minimum distance between p and some point g on the axis must then be the normal from p.
In modern mathematics, normals to curves are known for being the location of the center of curvature of that small part of the curve located around the foot. The distance from the foot to the center is the radius of curvature. The latter is the radius of a circle, but for other than circular curves, the small arc can be approximated by a circular arc. The curvature of non-circular curves; e.g., the conic sections, must change over the section. A map of the center of curvature; i.e., its locus, as the foot moves over the section, is termed the evolute of the section. Such a figure, the edge of the successive positions of a line, is termed an envelope today. Heath believed that in Book V we are seeing Apollonius establish the logical foundation of a theory of normals, evolutes, and envelopes.
Heath's was accepted as the authoritative interpretation of Book V for the entire 20th century, but the changing of the century brought with it a change of view. In 2001, Apollonius scholars Fried & Unguru, granting all due respect to other Heath chapters, balked at the historicity of Heath's analysis of Book V, asserting that he “reworks the original to make it more congenial to a modern mathematician ... this is the kind of thing that makes Heath’s work of dubious value for the historian, revealing more of Heath’s mind than that of Apollonius.” Some of his arguments are in summary as follows. There is no mention of maxima/minima being per se normals in either the prefaces or the books proper. Out of Heath's selection of 50 propositions said to cover normals, only 7, Book V: 27-33, state or imply maximum/minimum lines being perpendicular to the tangents. These 7 Fried classifies as isolated, unrelated to the main propositions of the book. They do not in any way imply that maxima/minima in general are normals. In his extensive investigation of the other 43 propositions, Fried proves that many cannot be.
Fried and Unguru counter by portraying Apollonius as a continuation of the past rather than a foreshadowing of the future. First is a complete philological study of all references to minimum and maximum lines, which uncovers a standard phraseology. There are three groups of 20-25 propositions each. The first group contains the phrase “from a point on the axis to the section,” which is exactly the opposite of a hypothetical “from a point on the section to the axis.” The former does not have to be normal to anything, although it might be. Given a fixed point on the axis, of all the lines connecting it to all the points of the section, one will be longest (maximum) and one shortest (minimum). Other phrases are “in a section,” “drawn from a section,” “cut off between the section and its axis,” cut off by the axis,” all referring to the same image.
In the view of Fried and Unguru, the topic of Book V is exactly what Apollonius says it is, maximum and minimum lines. These are not code words for future concepts, but refer to ancient concepts then in use. The authors cite Euclid, Elements, Book III, which concerns itself with circles, and maximum and minimum distances from interior points to the circumference. Without admitting to any specific generality they use terms such as “like” or “the analog of.” They are known for innovating the term “neusis-like.” A neusis construction was a method of fitting a given segment between two given curves. Given a point P, and a ruler with the segment marked off on it. one rotates the ruler around P cutting the two curves until the segment is fitted between them. In Book V, P is the point on the axis. Rotating a ruler around it, one discovers the distances to the section, from which the minimum and maximum can be discerned. The technique is not applied to the situation, so it is not neusis. The authors use neusis-like, seeing an archetypal similarity to the ancient method.
Book VI.
Book VI, known only through translation from the Arabic, contains 33 propositions, the least of any book. It also has large lacunae, or gaps in the text, due to damage or corruption in the previous texts.
The topic is relatively clear and uncontroversial. Preface 1 states that it is “equal and similar sections of cones.” Apollonius extends the concepts of congruence and similarity presented by Euclid for more elementary figures, such as triangles, quadrilaterals, to conic sections. Preface 6 mentions “sections and segments” that are “equal and unequal” as well as “similar and dissimilar,” and adds some constructional information.
Book VI features a return to the basic definitions at the front of the book. “Equality” is determined by an application of areas. If one figure; that is, a section or a segment, is “applied” to another (Halley's "si applicari possit altera super alteram"), they are “equal” (Halley's "aequales") if they coincide and no line of one crosses any line of the other. This is obviously a standard of congruence following Euclid, Book I, Common Notions, 4: “and things coinciding ("epharmazanta") with one another are equal ("isa").” Coincidence and equality overlap, but they are not the same: the application of areas used to define the sections depends on quantitative equality of areas but they can belong to different figures.
Between instances that are the same (homos), being equal to each other, and those that are different, or unequal, are figures that are “same-ish” (hom-oios), or similar. They are neither entirely the same nor different, but share aspects that are the same and do not share aspects that are different. Intuitively the geometricians had scale in mind; e.g., a map is similar to a topographic region. Thus figures could have larger or smaller versions of themselves.
The aspects that are the same in similar figures depend on the figure. Book 6 of Euclid's Elements presents similar triangles as those that have the same corresponding angles. A triangle can thus have miniatures as small as you please, or giant versions, and still be “the same” triangle as the original.
In Apollonius' definitions at the beginning of Book VI, similar right cones have similar axial triangles. Similar sections and segments of sections are first of all in similar cones. In addition, for every abscissa of one must exist an abscissa in the other at the desired scale. Finally, abscissa and ordinate of one must be matched by coordinates of the same ratio of ordinate to abscissa as the other. The total effect is as though the section or segment were moved up and down the cone to achieve a different scale.
Book VII.
Book VII, also a translation from the Arabic, contains 51 Propositions. These are the last that Heath considers in his 1896 edition. In Preface I, Apollonius does not mention them, implying that, at the time of the first draft, they may not have existed in sufficiently coherent form to describe. Apollonius uses obscure language, that they are “peri dioristikon theorematon”, which Halley translated as “de theorematis ad determinationem pertinentibus,” and Heath as “theorems involving determinations of limits.” This is the language of definition, but no definitions are forthcoming. Whether the reference might be to a specific kind of definition is a consideration but to date nothing credible has been proposed. The topic of Book VII, completed toward the end of Apollonius’ life and career, is stated in Preface VII to be diameters and “the figures described upon them,” which must include conjugate diameters, as he relies heavily on them. In what way the term “limits” or “determinations” might apply is not mentioned.
Diameters and their conjugates are defined in Book I (Definitions 4-6). Not every diameter has a conjugate. The topography of a diameter (Greek diametros) requires a regular curved figure. Irregularly-shaped areas, addressed in modern times, are not in the ancient game plan. Apollonius has in mind, of course, the conic sections, which he describes in often convolute language: “a curve in the same plane” is a circle, ellipse or parabola, while “two curves in the same plane” is a hyperbola. A chord is a straight line whose two end points are on the figure; i.e., it cuts the figure in two places. If a grid of parallel chords is imposed on the figure, then the diameter is defined as the line bisecting all the chords, reaching the curve itself at a point called the vertex. There is no requirement for a closed figure; e.g., a parabola has a diameter.
A parabola has symmetry in one dimension. If you imagine it folded on its one diameter, the two halves are congruent, or fit over each other. The same may be said of one branch of a hyperbola. Conjugate diameters (Greek suzugeis diametroi, where suzugeis is “yoked together”), however, are symmetric in two dimensions. The figures to which they apply require also an areal center (Greek kentron), today called a centroid, serving as a center of symmetry in two directions. These figures are the circle, ellipse, and two-branched hyperbola. There is only one centroid, which must not be confused with the foci. A diameter is a chord passing through the centroid, which always bisects it.
For the circle and ellipse, let a grid of parallel chords be superimposed over the figure such that the longest is a diameter and the others are successively shorter until the last is not a chord, but is a tangent point. The tangent must be parallel to the diameter. A conjugate diameter bisects the chords, being placed between the centroid and the tangent point. Moreover, both diameters are conjugate to each other, being called a conjugate pair. It is obvious that any conjugate pair of a circle are perpendicular to each other, but in an ellipse, only the major and minor axes are, the elongation destroying the perpendicularity in all other cases.
Conjugates are defined for the two branches of a hyperbola resulting from the cutting of a double cone by a single plane. They are called conjugate branches. They have the same diameter. Its centroid bisects the segment between vertices. There is room for one more diameter-like line: let a grid of lines parallel to the diameter cut both branches of the hyperbola. These lines are chord-like except that they do not terminate on the same continuous curve. A conjugate diameter can be drawn from the centroid to bisect the chord-like lines.
These concepts mainly from Book I get us started on the 51 propositions of Book VII defining in detail the relationships between sections, diameters, and conjugate diameters. As with some of Apollonius other specialized topics, their utility today compared to Analytic Geometry remains to be seen, although he affirms in Preface VII that they are both useful and innovative; i.e., he takes the credit for them.
Lost and reconstructed works described by Pappus.
Pappus mentions other treatises of Apollonius:
Each of these was divided into two books, and—with the "Data", the "Porisms", and "Surface-Loci" of Euclid and the "Conics" of Apollonius—were, according to Pappus, included in the body of the ancient analysis. Descriptions follow of the six works mentioned above.
"De Rationis Sectione".
"De Rationis Sectione" sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.
"De Spatii Sectione".
"De Spatii Sectione" discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.
In the late 17th century, Edward Bernard discovered a version of "De Rationis Sectione" in the Bodleian Library. Although he began a translation, it was Halley who finished it and included it in a 1706 volume with his restoration of "De Spatii Sectione".
"De Sectione Determinata".
"De Sectione Determinata" deals with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either (1) to the square on the remaining one or the rectangle contained by the remaining two or (2) to the rectangle contained by the remaining one and another given straight line. Several have tried to restore the text to discover Apollonius's solution, among them Snellius (Willebrord Snell, Leiden, 1698); Alexander Anderson of Aberdeen, in the supplement to his "Apollonius Redivivus" (Paris, 1612); and Robert Simson in his "Opera quaedam reliqua" (Glasgow, 1776), by far the best attempt.
"De Tactionibus".
"De Tactionibus" embraced the following general problem: Given three things (points, straight lines, or circles) in position, describe a circle passing through the given points and touching the given straight lines or circles. The most difficult and historically interesting case arises when the three given things are circles. In the 16th century, Vieta presented this problem (sometimes known as the Apollonian Problem) to Adrianus Romanus, who solved it with a hyperbola. Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work "Apollonius Gallus" (Paris, 1600). The history of the problem is explored in fascinating detail in the preface to J. W. Camerer's brief "Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c" (Gothae, 1795, 8vo).
"De Inclinationibus".
The object of "De Inclinationibus" was to demonstrate how a straight line of a given length, tending towards a given point, could be inserted between two given (straight or circular) lines. Though Marin Getaldić and Hugo d'Omerique ("Geometrical Analysis", Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).
"De Locis Planis".
"De Locis Planis" is a collection of propositions relating to loci that are either straight lines or circles. Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat ("Oeuvres", i., 1891, pp. 3–51) and F. Schooten (Leiden, 1656) but also, most successfully of all, by R. Simson (Glasgow, 1749).
Lost works mentioned by other ancient writers.
Ancient writers refer to other works of Apollonius that are no longer extant:
Early printed editions.
The early printed editions began for the most part in the 16th century. At that time, scholarly books were expected to be in Latin, today's New Latin. As almost no manuscripts were in Latin, the editors of the early printed works translated from the Greek or Arabic to Latin. The Greek and Latin were typically juxtaposed, but only the Greek is original, or else was restored by the editor to what he thought was original. Critical apparatuses were in Latin. The ancient commentaries, however, were in ancient or medieval Greek. Only in the 18th and 19th centuries did modern languages begin to appear. A representative list of early printed editions is given below. The originals of these printings are rare and expensive. For modern editions in modern languages see the references.
Ideas attributed to Apollonius by other writers.
Apollonius' contribution to astronomy.
The equivalence of two descriptions of planet motions, one using excentrics and another deferent and epicycles, is attributed to him. Ptolemy describes this equivalence as Apollonius' theorem in the "Almagest" XII.1.
Methods of Apollonius.
According to Heath, “The Methods of Apollonius” were not his and were not personal. Whatever influence he had on later theorists was that of geometry, not of his own innovation of technique. Heath says,
As a preliminary to the consideration in detail of the methods employed in the Conics, it may be stated generally that they follow steadily the accepted principles of geometrical investigation which found their definitive expression in the Elements of Euclid.
With regard to moderns speaking of golden age geometers, the term "method" means specifically the visual, reconstructive way in which the geometer unknowingly produces the same result as an algebraic method used today. As a simple example, algebra finds the area of a square by squaring its side. The geometric method of accomplishing the same result is to construct a visual square. Geometric methods in the golden age could produce most of the results of elementary algebra.
Geometrical algebra.
Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ...”
For example, in Apollonius a line segment AB (the line between Point A and Point B) is also the numerical length of the segment. It can have any length. AB therefore becomes the same as an algebraic variable, such as "x" (the unknown), to which any value might be assigned; e.g., "x"=3.
Variables are defined in Apollonius by such word statements as “let AB be the distance from any point on the section to the diameter,” a practice that continues in algebra today. Every student of basic algebra must learn to convert “word problems” to algebraic variables and equations, to which the rules of algebra apply in solving for "x". Apollonius had no such rules. His solutions are geometric.
Relationships not readily amenable to pictorial solutions were beyond his grasp; however, his repertory of pictorial solutions came from a pool of complex geometric solutions generally not known (or required) today. One well-known exception is the indispensable Pythagorean Theorem, even now represented by a right triangle with squares on its sides illustrating an expression such as a2 + b2 = c2. The Greek geometers called those terms “the square on AB,” etc. Similarly, the area of a rectangle formed by AB and CD was "the rectangle on AB and CD."
These concepts gave the Greek geometers algebraic access to linear functions and quadratic functions, which latter the conic sections are. They contain powers of 1 or 2 respectively. Apollonius had not much use for cubes (featured in solid geometry), even though a cone is a solid. His interest was in conic sections, which are plane figures. Powers of 4 and up were beyond visualization, requiring a degree of abstraction not available in geometry, but ready at hand in algebra.
The coordinate system of Apollonius.
All ordinary measurement of length in public units, such as inches, using standard public devices, such as a ruler, implies public recognition of a Cartesian grid; that is, a surface divided into unit squares, such as one square inch, and a space divided into unit cubes, such as one cubic inch. The ancient Greek units of measurement had provided such a grid to Greek mathematicians since the Bronze Age.
Prior to Apollonius, Menaechmus and Archimedes had already started locating their figures on an implied window of the common grid by referring to distances conceived to be measured from a left-hand vertical line marking a low measure and a bottom horizontal line marking a low measure, the directions being rectilinear, or perpendicular to one another. These edges of the window become, in the Cartesian coordinate system, the axes. One specifies the rectilinear distances of any point from the axes as the coordinates. The ancient Greeks did not have that convention. They simply referred to distances.
Apollonius does have a standard window in which he places his figures. Vertical measurement is from a horizontal line he calls the “diameter.” The word is the same in Greek as it is in English, but the Greek is somewhat wider in its comprehension. If the figure of the conic section is cut by a grid of parallel lines, the diameter bisects all the line segments included between the branches of the figure. It must pass through the vertex (koruphe, "crown"). A diameter thus comprises open figures such as a parabola as well as closed, such as a circle. There is no specification that the diameter must be perpendicular to the parallel lines, but Apollonius uses only rectilinear ones.
The rectilinear distance from a point on the section to the diameter is termed tetagmenos in Greek, etymologically simply “extended.” As it is only ever extended “down” (kata-) or “up” (ana-), the translators interpret it as ordinate. In that case the diameter becomes the x-axis and the vertex the origin. The y-axis then becomes a tangent to the curve at the vertex. The abscissa is then defined as the segment of the diameter between the ordinate and the vertex.
Using his version of a coordinate system, Apollonius manages to develop in pictorial form the geometric equivalents of the equations for the conic sections, which raises the question of whether his coordinate system can be considered Cartesian. There are some differences. The Cartesian system is to be regarded as universal, covering all figures in all space applied before any calculation is done. It has four quadrants divided by the two crossed axes. Three of the quadrants include negative coordinates meaning directions opposite the reference axes of zero.
Apollonius has no negative numbers, does not explicitly have a number for zero, and does not develop the coordinate system independently of the conic sections. He works essentially only in Quadrant 1, all positive coordinates. Carl Boyer, a modern historian of mathematics, therefore says:
However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed "a posteriori" upon a given curve in order to study its properties ... Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry...
No one denies, however, that Apollonius occupies some sort of intermediate niche between the grid system of conventional measurement and the fully developed Cartesian Coordinate System of Analytic Geometry. In reading Apollonius, one must take care not to assume modern meanings for his terms.
The theory of proportions.
Apollonius uses the "Theory of Proportions" as expressed in Euclid’s "Elements", Books 5 and 6. Devised by Eudoxus of Cnidus, the theory is intermediate between purely graphic methods and modern number theory. A standard decimal number system is lacking, as is a standard treatment of fractions. The propositions, however, express in words rules for manipulating fractions in arithmetic. Heath proposes that they stand in place of multiplication and division.
By the term “magnitude” Eudoxus hoped to go beyond numbers to a general sense of size, a meaning it still retains. With regard to the figures of Euclid, it most often means numbers, which was the Pythagorean approach. Pythagoras believed the universe could be characterized by quantities, which belief has become the current scientific dogma. Book V of Euclid begins by insisting that a magnitude (megethos, “size”) must be divisible evenly into units (meros, “part”). A magnitude is thus a multiple of units. They do not have to be standard measurement units, such as meters or feet. One unit can be any designated line segment.
There follows perhaps the most useful fundamental definition ever devised in science: the ratio (Greek logos, meaning roughly “explanation.”) is a statement of relative magnitude. Given two magnitudes, say of segments AB and CD. the ratio of AB to CD, where CD is considered unit, is the number of CD in AB; for example, 3 parts of 4, or 60 parts per million, where ppm still uses the “parts” terminology. The ratio is the basis of the modern fraction, which also still means “part,” or “fragment”, from the same Latin root as fracture.
The ratio is the basis of mathematical prediction in the logical structure called a “proportion” (Greek analogos). The proportion states that if two segments, AB and CD, have the same ratio as two others, EF and GH, then AB and CD are proportional to EF and GH, or, as would be said in Euclid, AB is to CD as EF is to GH.
Algebra reduces this general concept to the expression AB/CD = EF/GH. Given any three of the terms, one can calculate the fourth as an unknown. Rearranging the above equation, one obtains AB = (CD/GH)•EF, in which, expressed as y = kx, the CD/GH is known as the “constant of proportionality.” The Greeks had little difficulty with taking multiples (Greek pollaplasiein), probably by successive addition.
Apollonius uses ratios almost exclusively of line segments and areas, which are designated by squares and rectangles. The translators have undertaken to use the colon notation introduced by Gottfried Wilhelm Leibniz in "Acta Eruditorum", 1684. Here is an example from "Conics", Book I, on Proposition 11:
External links.
Many of the popular sites in the history of mathematics linked below reference or analyze concepts attributed to Apollonius in modern notations and concepts. Since much of Apollonius is subject to interpretation, and he does not per se use modern vocabulary or concepts, the analyses below may not be optimal or accurate. They represent the historical theories of their authors.


Estimation of covariance matrices

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R"p"×"p"; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has normal distribution, the sample covariance matrix has Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.
Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles:
Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample.
Estimation in a general context.
Given a sample consisting of "n" independent observations "x"1..., "x""n" of a "p"-dimensional random vector "X" ∈ R"p"×1 (a "p"×1 column-vector), an unbiased estimator of the ("p"×"p") covariance matrix
is the sample covariance matrix
where formula_3 is the "i"-th observation of the "p"-dimensional random vector, and the vector
is the sample mean.
This is true regardless of the distribution of the random variable "X", provided of course that the theoretical means and covariances exist. The reason for the factor "n" − 1 rather than "n" is essentially the same as the reason for the same factor appearing in unbiased estimates of sample variances and sample covariances, which relates to the fact that the mean is not known and is replaced by the sample mean (see Bessel's correction).
In cases where the distribution of the random variable "X" is known to be within a certain family of distributions, other estimates may be derived on the basis of that assumption. A well-known instance is when the random variable "X" is normally distributed: in this case the maximum likelihood estimator of the covariance matrix is slightly different from the unbiased estimate, and is given by
A derivation of this result is given below. Clearly, the difference between the unbiased estimator and the maximum likelihood estimator diminishes for large "n".
In the general case, the unbiased estimate of the covariance matrix provides an acceptable estimate when the data vectors in the observed data set are all complete: that is they contain no missing elements. One approach to estimating the covariance matrix is to treat the estimation of each variance or pairwise covariance separately, and to use all the observations for which both variables have valid values. Assuming the missing data are missing at random this results in an estimate for the covariance matrix which is unbiased. However, for many applications this estimate may not be acceptable because the estimated covariance matrix is not guaranteed to be positive semi-definite. This could lead to estimated correlations having absolute values which are greater than one, and/or a non-invertible covariance matrix.
When estimating the cross-covariance of a pair of signals that are wide-sense stationary, missing samples do "not" need be random (e.g., sub-sampling by an arbitrary factor is valid).
Maximum-likelihood estimation for the multivariate normal distribution.
A random vector "X" ∈ R"p" (a "p"×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R"p" × "p" is a positive-definite matrix and the probability density function of "X" is
where "μ" ∈ R"p"×1 is the expected value of "X". The covariance matrix Σ is the multidimensional analog of what in one dimension would be the variance, and
normalizes the density formula_8 so that it integrates to 1.
Suppose now that "X"1, ..., "X""n" are independent and identically distributed samples from the distribution above. Based on the observed values "x"1, ..., "x""n" of this sample, we wish to estimate Σ.
First steps.
The likelihood function is:
It is fairly readily shown that the maximum-likelihood estimate of the mean vector "μ" is the "sample mean" vector:
See the section on estimation in the article on the normal distribution for details; the process here is similar.
Since the estimate formula_11 does not depend on Σ, we can just substitute it for "μ" in the likelihood function, getting
and then seek the value of Σ that maximizes the likelihood of the data (in practice it is easier to work with log formula_13).
The trace of a 1 × 1 matrix.
Now we come to the first surprising step: regard the scalar formula_14 as the trace of a 1×1 matrix. This makes it possible to use the identity tr("AB") = tr("BA") whenever "A" and "B" are matrices so shaped that both products exist. We get
where
formula_17 is sometimes called the scatter matrix, and is positive definite if there exists a subset of the data consisting of formula_18 affinely independent observations (which we will assume).
Using the spectral theorem.
It follows from the spectral theorem of linear algebra that a positive-definite symmetric matrix "S" has a unique positive-definite symmetric square root "S"1/2. We can again use the "cyclic property" of the trace to write
Let "B" = "S"1/2 Σ −1 "S"1/2. Then the expression above becomes
The positive-definite matrix "B" can be diagonalized, and then the problem of finding the value of "B" that maximizes
Since the trace of a square matrix equals the sum of eigen-values ("trace and eigenvalues"), the equation reduces to the problem of finding the eigen values λ1, ..., λ"p" that maximize
This is just a calculus problem and we get λ"i" = "n" for all "i." Thus, assume "Q" is the matrix of eigen vectors, then
i.e., "n" times the "p"×"p" identity matrix.
Concluding steps.
Finally we get
i.e., the "p"×"p" "sample covariance matrix"
is the maximum-likelihood estimator of the "population covariance matrix" Σ. At this point we are using a capital "X" rather than a lower-case "x" because we are thinking of it "as an estimator rather than as an estimate", i.e., as something random whose probability distribution we could profit by knowing. The random matrix "S" can be shown to have a Wishart distribution with "n" − 1 degrees of freedom. That is:
Alternative derivation.
An alternative derivation of the maximum likelihood estimator can be performed via matrix calculus formulae (see also differential of a determinant and differential of the inverse matrix). It also verifies the aforementioned fact about the maximum likelihood estimate of the mean. Re-write the likelihood in the log form using the trace trick:
The differential of this log-likelihood is
It naturally breaks down into the part related to the estimation of the mean, and to the part related to the estimation of the variance. The first order condition for maximum, formula_29, is satisfied when the terms multiplying formula_30 and formula_31 are identically zero. Assuming (the maximum likelihood estimate of) formula_32 is non-singular, the first order condition for the estimate of the mean vector is
which leads to the maximum likelihood estimator
This lets us simplify
as defined above. Then the terms involving formula_31 in formula_37 can be combined as
The first order condition formula_29 will hold when the term in the square bracket is (matrix-valued) zero. Pre-multiplying the latter by formula_32 and dividing by formula_41 gives
which of course coincides with the canonical derivation given earlier.
Dwyer points out that decomposition into two terms such as appears above is "unnecessary" and derives the estimator in two lines of working. Note that it may be not trivial to show that such derived estimator is the unique global maximizer for likelihood function.
Intrinsic covariance matrix estimation.
Intrinsic expectation.
Given a sample of "n" independent observations "x"1..., "x""n" of a "p"-dimensional zero-mean Gaussian random variable "X" with covariance R, the maximum likelihood estimator of R is given by
The parameter R belongs to the set of positive-definite matrices, which is a Riemannian manifold, not a vector space, hence the usual vector-space notions of expectation, i.e. "E[R^]", and estimator bias must be generalized to manifolds to make sense of the problem of covariance matrix estimation. This can be done by defining the expectation of an manifold-valued estimator R^ with respect to the manifold-valued point R as
where
are the exponential map and inverse exponential map, respectively, "exp" and "log" denote the ordinary matrix exponential and matrix logarithm, and E[·] is the ordinary expectation operator defined on a vector space, in this case the tangent space of the manifold.
Bias of the sample covariance matrix.
The intrinsic bias vector field of the SCM estimator formula_47 is defined to be
The intrinsic estimator bias is then given by formula_49.
For complex Gaussian random variables, this bias vector field can be shown to equal
where
and ψ(·) is the digamma function. The intrinsic bias of the sample covariance matrix equals
and the SCM is asymptotically unbiased as "n" → ∞.
Similarly, the intrinsic inefficiency of the sample covariance matrix depends upon the Riemannian curvature of the space of positive-definite matrices.
Shrinkage estimation.
If the sample size "n" is small and the number of considered variables "p" is large, the above empirical estimators of covariance and correlation are very unstable. Specifically, it is possible to furnish estimators that improve considerably upon the maximum likelihood estimate in terms of mean squared error. Moreover, for "n" < "p" (the number of observations is less than the number of random variables) the empirical estimate of the covariance matrix becomes singular, i.e. it cannot be inverted to compute the precision matrix.
As an alternative, many methods have been suggested to improve the estimation of the covariance matrix. All of these approaches rely on the concept of shrinkage. This is implicit in Bayesian methods and in penalized maximum likelihood methods and explicit in the Stein-type shrinkage approach.
A simple version of a shrinkage estimator of the covariance matrix is represented by the Ledoit-Wolf shrinkage estimator. One considers a convex combination of the empirical estimator (formula_53) with some suitable chosen target (formula_54), e.g., the diagonal matrix. Subsequently, the mixing parameter (formula_55) is selected to maximize the expected accuracy of the shrunken estimator. This can be done by cross-validation, or by using an analytic estimate of the shrinkage intensity. The resulting regularized estimator (formula_56) can be shown to outperform the maximum likelihood estimator for small samples. For large samples, the shrinkage intensity will reduce to zero, hence in this case the shrinkage estimator will be identical to the empirical estimator. Apart from increased efficiency the shrinkage estimate has the additional advantage that it is always positive definite and well conditioned.
Various shrinkage targets have been proposed:
The shrinkage estimator can be generalized to a multi-target shrinkage estimator that utilizes several targets simultaneously. Software for computing a covariance shrinkage estimator is available in R (packages corpcor and ShrinkCovMat), in Python (library scikit-learn), and in MATLAB.


Magnetic level gauge

A magnetic level gauge is a type of level sensor, i.e., a device used to measure the level of fluids. A magnetic level gauge includes a “floatable” device that can float both in high and low density fluids. Magnetic level gauges may also be designed to accommodate severe environmental conditions up to 210 bars at 370 °C.
Magnetic level sensors.
Magnetic float level sensors involve the use of a permanent magnet sealed inside a float whose rise and fall causes the opening or closing of a mechanical switch, either through direct contact or in proximity of a reed switch. With mechanically actuated floats, the float is directly connected to a micro switch. For both magnetic and mechanical float level sensors, chemical compatibility, temperature, specific gravity (density), buoyancy, and viscosity affect the selection of the stem and the float. For example, larger floats may be used with liquids with specific gravities as low as 0.5 while still maintaining buoyancy. The choice of float material is also influenced by temperature-induced changes in specific gravity and viscosity – changes that directly affect buoyancy.
Explanation.
Exploring the physics and engineering behind this design, requires looking at basic magnetism. A standard bar magnet has two magnetic poles: north and south. (The north will read positive on a gauss meter and the south will read negative.) Magnetic field are mapped using magnetic flux lines. These lines are a graphical representation of the magnetic field density. They show the direction of flow for the magnetic field and represent relative field strength – the closer together the lines are, the stronger the magnetic field. Flux lines will always travel from the north pole to the nearest south pole and always leave and enter surfaces at 90°, or perpendicular to the surface. They can only travel in straight lines or curved paths, which means they can never make a sudden, abrupt change in direction. Flux lines will also always follow a path of least magnetic resistance. Most importantly, they can never cross one another.
Considerations.
When selecting a magnetic level gauge it is important to take into account the strength of the magnetic field. The magnetic field is the heart of the magnetic level gauge – the stronger the field, the more reliable the instrument will function. Some manufacturers rely on a single magnet for their magnetic level gauges which causes the strength of the north field to be identical to, and as weak as, the south field. It is apparent that at the location of the indicators, switches and transmitters, the field would not be as intense. Some manufacturers use a single annular ring magnet, others use a series of single bar magnets in a circular array in their float design. In this design the relative field strength of the north and south poles will be equal to one another and less than that of a dual magnet design. Moreover, the field strength as you travel around the circumference will have high and low spots as you pass between the individual bar magnets.


Mathematical principles of reinforcement

The mathematical principles of reinforcement (MPR) constitute of a set of mathematical equations set forth by Peter Killeen and his colleagues attempting to describe and predict the most fundamental aspects of behavior (Killeen & Sitomer, 2003).
The three key principles of MPR, arousal, constraint, and coupling, describe how incentives motivate responding, how time constrains it, and how reinforcers become associated with specific responses, respectively. Mathematical models are provided for these basic principles in order to articulate the necessary detail of actual data.
First principle: arousal.
The first basic principle of MPR is arousal. Arousal refers to the activation of behavior by the presentation of incentives. An increase in activity level following repeated presentations of incentives is a fundamental aspect of conditioning. Killeen, Hanson, and Osborne (1978) proposed that adjunctive (or schedule induced) behaviors are normally occurring parts of an organism's repertoire. Delivery of incentives increases the rate of adjunctive behaviors by generating a heightened level of general activity, or arousal, in organisms.
Killeen & Hanson (1978) exposed pigeons to a single daily presentation of food in the experimental chamber and measured general activity for 15 minutes after a feeding. They showed that activity level increased slightly directly following a feeding and then decreased slowly over time. The rate of decay can be described by the following function:
The time course of the entire theoretical model of general activity is modeled by the following equation:
To better conceptualize this model, imagine how rate of responding would appear with each of these processes individually. In the absence of temporal inhibition or competing responses, arousal level would remain high and response rate would be depicted as an almost horizontal line with a very small negative slope. Directly following food presentation, temporal inhibition is at its maximum level. It decreases quickly as time elapses, and response rate would be expected to increase up to the level of arousal in a short time. Competing behaviors such as goal tracking or hopper inspection are at a minimum directly after food presentation. These behaviors increase as the interval elapses, so the measure of general activity would slowly decrease. Subtracting these two curves results in the predicted level of general activity.
Killeen et al. (1978) then increased the frequency of feeding from daily to every fixed-time seconds. They showed that general activity level increased substantially from the level of daily presentation. Response rate asymptotes were highest for the highest rates of reinforcement. These experiments indicate that arousal level is proportional to rate of incitement, and the asymptotic level increases with repeated presentations of incentives. The increase in activity level with repeated presentation of incentives is called cumulation of arousal. The first principle of MPR states that arousal level is proportional to rate of reinforcement, formula_4, where:
= arousal level
= specific activation
= rate of reinforcement
(Killeen & Sitomer, 2003).
Second principle: constraint.
An obvious but often overlooked factor when analyzing response distributions is that responses are not instantaneous, but take some amount of time to emit (Killeen, 1994). These ceilings on response rate are often accounted for by competition from other responses, but less often for the fact that responses cannot always be emitted at the same rate at which they are elicited (Killeen & Sitomer, 2003). This limiting factor must be taken into account in order to correctly characterize what responding could be theoretically, and what it will be empirically.
An organism may receive impulses to respond at a certain rate. At low rates of reinforcement, the elicited rate and emitted rate will approximate each other. At high rates of reinforcement, however, this elicited rate is subdued by the amount of time it takes to emit a response. Response rate, formula_5, is typically measured as the number of responses occurring in an epoch divided by the duration of an epoch. The reciprocal of formula_5 gives the typical measure of the inter response (IRT), the average time from the start of one response to the start of another (Killeen & Sitomer, 2003). This is actually the cycle time rather than the time between responses. According to Killeen & Sitomer (2003), the IRT consists of two subintervals, the time required to emit a response, formula_7 plus the time between responses, formula_8. Therefore, response rate can be measured either by dividing the number of responses by the cycle time:
or as the number of responses divided by the actual time between responses:
This instantaneous rate, formula_11 may be the best measure to use, as the nature of the operandum may change arbitrarily within an experiment (Killeen & Sitomer, 2003).
Killeen, Hall, Reilly, and Kettle (2002) showed that if instantaneous rate of responding is proportional to rate of reinforcement, formula_12, then a fundamental equation for MPR results. Killeen & Sitomer (2003) showed that:
if formula_13
then formula_14,
and rearranging gives:
formula_15
While responses may be elicited at a rate proportional to formula_16, they can only be emitted at rate formula_5 due to constraint. The second principle of MPR states that the time required to emit a response constrains response rate (Killeen & Sitomer, 2003).
Third principle: coupling.
Coupling is the final concept of MPR that ties all of the processes together and allows for specific predictions of behavior with different schedules of reinforcement. Coupling refers to the association between responses and reinforcers. The target response is the response of interest to the experimenter, but any response can become associated with a reinforcer. Contingencies of reinforcement refer to how a reinforcer is scheduled with respect to the target response (Killeen & Sitomer, 2003), and the specific schedules of reinforcement in effect determine how responses are coupled to the reinforcer. The third principle of MPR states that the degree of coupling between a response and reinforcer decreases with the distance between them (Killeen & Sitomer, 2003). Coupling coefficients, designated as formula_18, are given for the different schedules of reinforcement. When the coupling coefficients are inserted into the activation-constraint model, complete models of conditioning are derived:
This is the fundamental equation of MPR. The dot after the formula_20 is a placeholder for the specific contingencies of reinforcement under study (Killeen & Sitomer, 2003).
Fixed-ratio reinforcement schedules.
The rate of reinforcement for fixed-ratio schedules is easy to calculate, as reinforcement rate is directly proportional to response rate and inversely proportional to ratio requirement (Killeen, 1994). The schedule feedback function is therefore:
Substituting this function into the complete model gives the equation of motion for ratio schedules (Killeen & Sitomer, 2003). Killeen (1994, 2003) showed that the most recent response in a sequence of responses is weighted most heavily and given a weight of formula_22, leaving formula_23 for the remaining responses. The penultimate response receives formula_24, the third back receives formula_25. The formula_26th response back is given a weight of formula_27
The sum of this series is the coupling coefficient for fixed-ratio schedules:
The continuous approximation of this is:
where formula_30 is the intrinsic rate of memory decay. Inserting the reinforcement rate and coupling coefficient into the activation-constraint model gives the predicted response rates for FR schedules:
This equation predicts low response rates at low ratio requirements due to the displacement of memory by consummatory behavior. However, these low rates are not always found. Coupling of responses may extend back beyond the preceding reinforcer, and an extra parameter, formula_32 is added to account for this. Killeen & Sitomer (2003) showed that the coupling coefficient for FR schedules then becomes:
formula_32 is the number of responses preceding the prior reinforcer that contribute to response strength. formula_35 which ranges from 0 to 1 is then the degree of erasure of the target response from memory with the delivery of a reinforcer. (formula_36) If formula_37, erasure is complete and the simpler FR equation can be used.
Variable-ratio reinforcement schedules.
According to Killeen & Sitomer (2003), the duration of a response can affect the rate of memory decay. When response durations vary, either within or between organisms, then a more complete model is needed, and formula_22 is replaced with formula_39 yielding:
Idealized variable-ratio schedules with a mean response requirement of formula_26 have a constant probability of formula_42 of a response ending in reinforcement (Bizo, Kettle, & Killeen, 2001). The last response ending in reinforcement must always occur and receives strengthening of formula_22. The penultimate response occurs with probability formula_44 and receives a strengthening of formula_24 . The sum of this process up to infinity is (Killeen 2001, Appendix):
The coupling coefficient for VR schedules ends up being:
formula_47
Multiplying by degree of erasure of memory gives:
formula_48
The coupling coefficient can then be inserted into the activation-constraint model just as the coupling coefficient for FR schedules to yield predicted response rates under VR schedules:
formula_49
In interval schedules, the schedule feedback function is
formula_50
where formula_51 is the minimum average time between reinforcers (Killeen, 1994). Coupling in interval schedules is weaker than ratio schedules, as interval schedules equally strengthen all responses preceding the target rather than just the target response. Only some proportion formula_52 of memory is strengthened. With a response requirement, the final, target response must receive strength of formula_22. All preceding responses, target or non-target, receive a strengthening of formula_23.
Fixed-time schedules are the simplest time dependent schedules in which organisms must simply wait t seconds for an incentive. Killeen (1994) reinterpreted temporal requirements as response requirements and integrated the contents of memory from one incentive to the next. This gives the contents of memory to be:
N
MN= lò e-lndn
0
This is the degree of saturation in memory of all responses, both target and non-target, elicited in the context (Killeen, 1994). Solving this equation gives the coupling coefficient for fixed-time schedules:
c=r(1-e-lbt)
where formula_52 is the proportion of target responses in the response trajectory. Expanding into a power series gives the following approximation:
c» rlbt
 1+lbt
This equation predicts serious instability for non-contingent schedules of reinforcement.
Fixed-interval schedules are guaranteed a strengthening of a target response, b=w1, as reinforcement is contingent on this final, contiguous response (Killeen, 1994). This coupling is equivalent to the coupling on FR 1 schedules
w1=b=1-e-l.
The remainder of coupling is due to the memory of preceding behavior. The coupling coefficient for FI schedules is:
c= b +r(1- b -e-lbt).
Variable-time schedules are similar to random ratio schedules in that there is a constant probability of reinforcement, but these reinforcers are set up in time rather than responses. The probability of no reinforcement occurring before some time t’ is an exponential function of that time with the time constant t being the average IRI of the schedule (Killeen, 1994). To derive the coupling coefficient, the probability of the schedule not having ended, weighted by the contents of memory, must be integrated.
M= lò e-n’t/te-ln’ dn’
 0
In this equation, t’=n’t, where t is a small unit of time. Killeen (1994) explains that the first exponential term is the reinforcement distribution, whereas the second term is the weighting of this distribution in memory. Solving this integral and multiplying by the coupling constant r, gives the extent to which memory is filled on VT schedules:
c=rlbt
 1+lbt
This is the same coupling coefficient as an FT schedule, except it is an exact solution for VT schedules rather than an approximation. Once again, the feedback function on these non-contingent schedules predicts serious instability in responding.
As with FI schedules, variable-interval schedules are guaranteed a target response coupling of b. Simply adding b to the VT equation gives:
M= b+ lò e-n’t/te-ln’ dn’
 1
Solving the integral and multiplying by r gives the coupling coefficient for VI schedules:
c= b+(1-b) rlbt
 1+lbt
The coupling coefficients for all of the schedules are inserted into the activation-constraint model to yield the predicted, overall response rate. The third principle of MPR states that the coupling between a response and a reinforcer decreases with increased time between them (Killeen & Sitomer, 2003).
Mathematical principles of reinforcement describe how incentives fuel behavior, how time constrains it, and how contingencies direct it. It is a general theory of reinforcement that combines both contiguity and correlation as explanatory processes of behavior. Many responses preceding reinforcement may become correlated with the reinforcer, but the final response receives the greatest weight in memory. Specific models are provided for the three basic principles to articulate predicted response patterns in many different situations and under different schedules of reinforcement. Coupling coefficients for each reinforcement schedule are derived and inserted into the fundamental equation to yield overall predicted response rates.


2396 Kochi


Mean integrated squared error

In statistics, the mean integrated squared error (MISE) is used in density estimation. The MISE of an estimate of an unknown probability density is given by
where "ƒ" is the unknown density, "ƒ""n" is its estimate based on a sample of "n" independent and identically distributed random variables.
Here, E denotes the expected value with respect to that sample.
The MISE is also known as "L"2 risk function.


Tachymeter (watch)

A tachymeter (pronounced ) is a scale sometimes inscribed around the rim of an analog watch with a chronograph. It can be used to conveniently compute the frequency in hours of an event of a known second-defined period, such as speed (distance over hours) based on travel time (seconds over distance), or measure distance based on speed. The spacings between the marks on the tachymeter dial are therefore proportional to , where "t" is the elapsed time.
The function performed by a tachymeter is independent of the unit of distance (e.g. statute miles, nautical miles, kilometres, metres etc.) as long as the same unit of length is used for all calculations. It can also be used to measure the frequency of any regular event in occurrences per hour, such as the units output by an industrial process. A tachymeter is simply a means of converting "elapsed time" (in seconds per unit) to "rate" (in units per hour).
Measuring speed.
To use a tachymeter-equipped watch for measuring speed, the chronograph is started at a starting marker of a known distance. At the next marker, the point on the scale adjacent to the second hand indicates the speed (in distance between markers per hour) of travel between the two. The typical tachymeter scale on a watch converts between the number of seconds it takes for an event to happen and the number of times that event will occur in one hour. The formula used to create this type of tachymeter scale is
where T is the tachymeter scale value; t is the time in seconds that it takes for the event to occur; and 3600 is the number of seconds in an hour.
As a sample calculation, if it takes 35 seconds to travel one mile, then the average speed is 103 miles/hour. On the watch, 35 seconds gives scale value 103. Similarly, if one kilometre takes 35 seconds then the average speed would be 103 km/hour.
Note that the tachymeter scale only calculates the average speed.
As a second example, if it takes 20 seconds to travel one unit of distance, then the average speed on the watch used for the purpose of the picture only is 180 units of distance per hour (examine the picture of the watch which is here to simplify the idea, actual tachymeters may vary slightly).
For events that happen either very quickly or slowly, one can adjust the sixty-second tachymeter scale commonly found on watches. Smaller fractional units can be used for slower objects, but the same X/hour function remains constant. The scale on a watch is only valid for things that happen in 60 seconds or fewer, and the scale is also difficult to resolve for events that take fewer than 10 seconds or so to occur. As an example, if it takes 100 seconds to eat an apple, cutting that number in half allows one to say that it takes 50 seconds to eat half an apple. Using the tachymeter scale one can calculate that 72 half apples (36 whole apples) could be eaten in one hour. Some watches, not common, have "wraparound" or "scroll" scales, which extend the readings to lower speeds, typically 45 units.
A counter-intuitive method of measuring speeds slower than can be directly indicated was published by Michael Bilow on the web site TimeZone.com: "Because the tachymeter scale is calibrated quite simply as divisors of 3600, it is easy to use the 0-60 segment as a reference scale for the reciprocals. That is, since something which traverses the unit distance in 45 seconds is going 80 units per hour, then it follows mathematically that something which traverses the unit distance in 80 seconds is going 45 units per hour. Since something which traverses the unit distance in 30 seconds is going 120 units per hour, then it follows that something which traverses the unit distance in 120 seconds is going 30 units per hour... Using this method, it is fairly simple to read by inspection nearly any slowly moving object down to the range of 8-12 units per hour." The method was later explained in detail: "It is not intuitive to use the tachymeter scale in this way because there will, in all likelihood, not be any hands pointing to the important places on it. Unlike the conventional use of the tachymeter scale to measure things which take 60 seconds or less to traverse the unit distance, you must mentally move from the number of seconds measured on the chronograph to the same number on the tachymeter scale, and only then read the number opposite on the dial. The ability of the tachymeter scale to be used 'backwards' like this, freely interchanging time and rate, is a direct consequence of the fact that it is created as a table of reciprocals in the first place."
Measuring distance.
A tachymeter-equipped watch can be used to measure distance by timing the travel over the distance while the speed is held constant. The tachymeter scale is rotated to align with the second hand at the start of the length to be measured. When the second hand reaches the point on the scale where the speed indicated equals the speed of the vehicle, one unit of distance (miles if speed is miles per hour, kilometres if kilometres per hour, etc.) has been covered. For example, if you travel at a constant 80 mph (or at 80 km/h), then the distance travelled while the second hand sweeps to "80" (45 seconds) will be exactly 1 mile (or 1 kilometre at 80 km/h).
Rotating scale.
Some tachymeter scales are on a rotating, indexed bezel. This allows two additional modes of use: The tachymeter bezel can be aligned with a free running second hand, and, more subtly, can be used to find the average speed over longer times/distances. Set the rotary bezel index to the position of the minute hand, note the current mileage/distance. Glance at the position of the minute hand on the tachymeter scale 60 units of distance later, and "average" speed will be indicated. A little mental math allows interim averages, easiest at 1/4 (15 unit) and other integer values. Alternatively, instead of using minute hand, align index bezel to the second hand and observe passing one unit of distance when position of the second hand will then indicate average speed.


222 Lucia

Lucia (minor planet designation: 222 Lucia) is a large Themistian asteroid. It was discovered by Johann Palisa on 9 February 1882 in Vienna and named after Lucia, daughter of Austro-Hungarian explorer Graf Wilczek.
This object is spectral C-type and is probably composed of primitive carbonaceous material. Based upon analysis of infrared spectra, it has a diameter of 59.8 ± 0.8 km. This object belongs to the Themis family, which was formed by the break-up of a larger parent body about a billion years ago.


Hook gauge evaporimeter

A Hook gauge evaporimeter is a precision instrument used to measure changes in water levels due to evaporation. It is used
to precisely measure the level of a free water surface as an evaporation pan or a tank. The main users are meteorologists and water engineers, especially in hot, arid countries where water conservation is of vital importance.
Description.
The device consists of a sharp hook suspended from a micrometer cylinder, with the body of the device having arms which rest on the rim of a still well inside the pan. The still well serves to isolate the device from any ripples that might be present in the sample being measured, while allowing the water level to equalize. The measurement is taken by turning the knob to lower the hook through the surface of the water until capillary action causes a small depression to form around the tip of the hook. The knob is then turned slowly until the depression "pops," with the measurement showing on the micrometer scale.
Use.
Evaporation rate is determined by a sequence of measurements over a set time interval, usually every 24 hours.


Web server benchmarking

Web server benchmarking is the process of estimating a web server performance in order to find if the server can serve sufficiently high workload.
Key parameters.
The performance is usually measured in terms of:
The measurements must be performed under a varying load of clients and requests per client.
Tools for benchmarking.
Load testing (stress/performance testing) a web server can be performed using automation/analysis tools such as:
Web application benchmarks.
Web application benchmarks measure the performance of application servers and database servers used to host web applications. TPC-W was a common benchmark emulating an online bookstore with synthetic workload generation.


Benjamin Richard Parker

Benjamin Richard Parker (often called Benjy by his sister) is a fictional character appearing in American comic books published by Marvel Comics. The character is depicted in the alternate future MC2 universe. He is the younger brother of May Parker / Spider-Girl, and the son of Mary Jane and Peter Parker / Spider-Man.
Fictional character biography.
Benjamin is born after a complicated pregnancy. Because his father's genetic code was altered when he received his powers, Ben is at a high risk of being born with some kind of genetic abnormality: deformity, disability, or perhaps even mutant powers. Due to the risk to Mary Jane's health, her obstetrician advises her to consider abortion. However, remembering that she had faced similar risks when pregnant with her daughter May, Mary Jane decides to proceed with the pregnancy.
Ben is born while his sister is fighting Seth, and to his family's relief, is apparently a perfectly healthy, normal little boy. His first name Benjamin is in honor of his uncle and great uncle, while his second name Richard is in honor of his grandfather.
Benjamin seems to display some superhuman abilities, namely balancing a toy block on one finger while spinning his arm at a fast speed. He was also seen dangling the block from his finger on a web-like string. May discovers Ben crawling on the ceiling of their home.
He was once possessed by a miniature version of the Carnage symbiote. After his sister freed him from the symbiote by using the ultrasonic weaponry of the villain Reverb, his father notices that the baby's ears were bleeding, and realized that Ben has lost his hearing, likely because Ben's ears were far too underdeveloped to withstand the sonic waves. The doctors in the hospital were trying to determine whether Ben's hearing loss is temporary or permanent. May is deeply upset over this and blames herself. Nevertheless, Ben still seems to be his usual, happy self. Since discovering her baby brother crawling on the ceiling she fears that due to his exposure to the symbiote, his abilities have somehow been jump started far too early (as hers only came about in her teens).
Normie Osborn agreed to fund an operation to restore Benjamin's hearing. The operation was a success, restoring most, if not all of Ben's hearing. When Benjy was going through it, Normie closely examined his body and found out that he was already developing his powers, but the exposure to the symbiote accelerated the process. Peter begins to fear that Benjy may be more powerful than him and his sister. This is strongly implied to be true when Ben spins organic webs to save both himself and Mary Jane after being thrown off a bridge by the Green Goblin, something neither Peter nor May can do. Despite being a baby, he was also strong enough for Mary Jane to hold on to without hurting him.
It is also known that Peter is the only one who can get him to burp "in the morning" (as Mary Jane says it) by feeding him chili.
Spider-Verse.
In the 2014/2015 crossover event "Spider-Verse", Benjy's family come under attack from Daemos, a relation of the 616 Spider-Man's former nemesis Morlun. During the attack, Mary Jane, Mayday's boyfriend Wes, and Peter are apparently all killed and their home destroyed. Mayday flees with Ben and is rescued by visiting Spider-Men from other dimensions who are trying to save as many Spiders as possible from similar attacks by Morlon and Daemos' family, who call themselves The Inheritors. Mayday and Ben are taken to a safe zone where the Spiders plan their next course of action. The safe zone is eventually compromised and Ben is captured by the Inheritors. It is revealed that the infant Ben is a vital part of a prophecy that will help bring about the downfall of the Inheritors and involves "The Other" (Kaine), "The Bride" (Silk) and "The Scion" (Ben himself). However, conversely if the three specific totems were sacrificed together, their deaths will ensure not only the Inheritors remain in power forever but also will stop future spider-people from appearing, and thus preventing the prophecy. Benjy is eventually saved by Ben Parker- his great-uncle and possible namesake- and Spider-Ham in the final fight, Uncle Ben taking Benjy to safety and Spider-Ham taking Benjy's place to catch the Inheritors off-guard. Afterwards, it is revealed Benjamin's mother and Wes survived the Inheritor attacked, but unfortunately his father did not.
Web Warriors.
Benjamin makes a few cameo appearances in 2015's Web Warriors series, looked after by Mayday, Mary Jane, Uncle Ben and is often visited by Anya Corzon.
Spider-Geddon.
Benjamin is referenced several times by his sister Mayday in the event Spider-Geddon. After the latest battle with The Inheritors has concluded, Mayday comments that her brother is very likely still the Scion of the Spider-Scroll Prophecy, her alternate world sister Annie May Parker, Spiderling, informs her that The Other is still in play too and is closer than she knows. On Mayday and Benjamin's Earth, it is revealed that The Other has resurrected their father.
Other versions.
Universe X.
On Earth X, when Spiders Man, an African American villain mutated by the Terrigen Mist, attacked a middle aged Peter using his web illusions and trapping Peter on a world of suffer. However, Peter took control of the illusion and fabricated a world where he had his ideal life, including his son Benjamin who had grown up and took the mantle of Spider-Man. Then Venom, Peter's real life daughter May who had bonded to the Venom symbiote, defeated Spiders Man and broke her out of that fabricated world and convinced Peter to accept his real life.
Arachnamorphosis.
A version of Benjamin appears in "What If?... Starring Spider-Man: Arachnamorphosis", in which he is the son of Peter Parker and Gwen Stacy. In this reality Spider-Man's powers began to mutate him into a monster along with Ben who inherited traits from the spider powers.
Peter and MJ happily ever after.
In this universe, Peter had overcome every hardship and he with his wife Mary Jane lived a good life, growing old along with their children, May and Ben.
Last Avenger.
In another universe, Ben was featured, where Spider-Man refused to join the Avengers.
Spider-Dreams.
Benjamin along with his family, May, Mary Jane and Peter, with the help of S.H.I.E.L.D. got new identities and moved away after Peter gave up being Spider-Man. He along with his family lived a full life and died of old age, while Peter, who gained longevity, outlived his own family.
Spider-Girl: The End.
In the alternate reality where April had accidentally killed her sister, April started impersonating her dead sister, but Benji saw through her disguise and revealed it MJ.
Adventures in Spidey-Baby-Sitting.
In this alternate universe, the son of Peter and MJ was born a spider-like human mutant, but despite of this, his parents still love him.
Spider-Man: Last Stand.
In the universe of "Spider-Man: Last Stand", Peter had a son named Ben.
House of M.
In the reality of House of M, Peter married Gwen and had a son named Richie.
Fantastic Four: The End.
In the reality of , the son of Peter wore his father's suit and under the alias of Spider-Kid joined the Insect Squad.


Geographical mile

The geographical mile is a unit of length determined by 1 minute of arc along the Earth's equator. For the international ellipsoid 1924 this equalled 1855.4 metres. "The American Practical Navigator" 2017 defines the geographical mile as . Greater precision depends more on choice of ellipsoid than on more careful measurement: the length of the equator in the World Geodetic System WGS-84 is which makes the geographical mile 1855.3248 m, while the "IERS Conventions (2010)" takes the equator to be making the geographical mile 1855.3250 m, 0.2 millimetres longer. In any ellipsoid, the length of a degree of longitude at the equator is thus exactly 60 geographical miles.
The shape of the Earth is a slightly flattened sphere, which results in the Earth's circumference being 0.168% larger when measured around the equator as compared to through the poles. The geographical mile is slightly larger than the nautical mile (which was historically linked to the circumference measured through both poles); one geographic mile is equivalent to approximately .
Related units.
It was closely related to the nautical mile, which was originally determined as 1 minute of arc along a great circle of the Earth, but is nowadays defined as exactly 1852 metres. The US National Institute of Standards and Technology (NIST) notes that: "The international nautical mile of 1 852 meters (6 076.115 49...feet) was adopted effective July 1, 1954, for use in the United States. The value formerly used in the United States was 6 080.20 feet = 1 nautical (geographical or sea) mile." (The deprecated value, 6080.2 feet, is n) A separate reference identifies the geographic mile as being identical to the international nautical mile of 1852 metres (and slightly shorter than the British nautical mile of ). The unit is not used much, but is cited in some United States laws (e.g., Section 1301(a) of the Submerged Lands Act, which defines state seaward boundaries in terms of geographic miles). While debating what became the Land Ordinance of 1785, Thomas Jefferson's committee wanted to divide the public lands in the west into "hundreds of ten geographical miles square, each mile containing 6086 and 4-10ths of a foot" and "sub-divided into lots of one mile square each, or 850 and 4-10ths of an acre".
The Danish and German geographical mile ("geografisk mil" and "geographische Meile" or "geographische Landmeile", respectively) is 4 minutes of arc, and was defined as approximately 7421.5 metres by the astronomer Ole Rømer of Denmark. In Norway and Sweden, this 4-minute geographical mile was mainly used at sea ("sjømil"), up to the beginning of the 20th century.


List of popular music songs featuring Andalusian cadences

Following is a list of popular music songs which feature a chord progression commonly known as Andalusian cadences.
Items in the list are sorted alphabetically by the band or artist's name. Songs which are familiar to listeners through more than one version (by different artists) are mentioned by the earliest version known to contain Andalusian cadences (which is most frequently the original version). Songs whose composers are unknown are at the bottom of the list, sorted alphabetically in their turn by title.


WebVR

WebVR was an experimental JavaScript application programming interface (API) that enabled applications to interact with virtual reality devices, such as the HTC Vive, Oculus Rift, Google Cardboard or Open Source Virtual Reality (OSVR) in a web browser. It was implemented in Firefox and Chromium-based browsers before being deprecated and removed. The WebVR API was superseded by a more powerful WebXR API that is capable of representing virtual reality and augmented reality devices simultaneously.
This API is designed with the following goals in mind:
History.
WebVR API was first conceived in spring 2014 by Vladimir Vukićević from Mozilla. The API's contributors include Brandon Jones, Boris Smus and others from the Mozilla team. On March 1, 2016, the Mozilla VR team and the Google Chrome team announced the version 1.0 release of the WebVR API proposal. The resulting API refactoring brought many improvements to WebVR.
The last tagged version is 1.1, which was last edited on April 5, 2017. The editors of the document include members from Mozilla and Google teams. However some Microsoft members have joined, and are actively collaborating in the drafting process of version 2.0 for the WebVR API.
Design.
The WebVR API exposes a few new interfaces (such as VR Display, VR pose) that allow web applications to present content in virtual reality, by using WebGL with the necessary camera settings and device interactions (such as controllers or point of view). The API has been designed to follow a certain path, which is very similar to other intrusive Web API like the Geolocation API. The necessary steps are:
Support.
WebVR 1.0 is currently supported in the release version of Firefox 55+ for Windows (64 bit version only) and was supported by Chrome for Android until version 80 as an Origin Trial Experiment, which means that developers can request a token to add to their website which will seamlessly enable WebVR. WebVR 1.1 is supported in Microsoft Edge from build 15002+, and in Samsung Internet, Chromium, Servo and Oculus Carmel.
Chrome for Windows supports WebVR 1.1 in a special build. Firefox 55+ on macOS also supports WebVR if a setting is enabled by the user.
Similar technologies.
Although WebVR is unique as an API, there are native applications on most hardware allowing for networked experiences and access to web content. Several key tools, such as Unity and Blender, are also able to export for the web and provide users a way to use their content without installing a dedicated application.
WebXR.
In 2018, the WebXR API superseded WebVR, being designed for both augmented reality and virtual reality devices.