Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier 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. With his predecessors Euclid and Archimedes, Apollonius is generally considered among the greatest mathematicians of antiquity.

Aside from geometry, Apollonius worked on numerous other topics, including astronomy. Most of this work has not survived, where exceptions are typically fragments referenced by other authors like Pappus of Alexandria. 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eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.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
Despite his momentous contributions to the field of mathematics, scant biographical information on  Apollonius remains. The 6th century Greek commentator Eutocius of Ascalon, writing on Apollonius' Conics, states:

Apollonius, the geometrician, ...  came from Perga in Pamphylia in the times of Ptolemy III Euergetes, so records Herakleios the biographer of Archimedes ....

From this passage Apollonius can be approximately dated, but specific birth and death years stated by modern scholars are only speculative. Ptolemy III Euergetes ("benefactor") was third Greek dynast of Egypt in the Diadochi succession, who reigned 246–222/221 BC. "Times" are always recorded by ruler or officiating magistrate, so Apollonius was likely born after 246. The identity of Herakleios is uncertain.

Perga was a Hellenized city in Pamphylia, Anatolia, whose ruins yet stand. It was a center of Hellenistic culture. 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 Attalids of Pergamon to the north. Someone designated "of Perga" might 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.

Autobiographical prefaces

Some autobiographical material can be found in the surviving prefaces to the books of Conics. These are letters Apollonius addressed to influential friends asking them to review the book enclosed with the letter. The first two prefaces are addressed to Eudemus of Pergamon.

Eudemus likely was or became the head of the research center of the  of Pergamon, a city known for its books and parchment industry from which the name parchment is derived. Research in Greek mathematical institutions, which followed the model of the Athenian Lycaeum, was part of the educational effort to which the library and museum were adjunct. There was only one such school in the state, under royal patronage. Books were rare and expensive and collecting them was a royal obligation.

Apollonius's preface to Book I tells Eudemus that the first four books were concerned with the development of elements while the last four were concerned with special topics. Apollonius reminds Eudemus that Conics was initially requested by Naucrates, a geometer and house guest at Alexandria otherwise unknown to history. Apollonius provided Naucrates the first draft of all eight books, but he refers to them as being "without a thorough purgation", and intended to verify and correct the books, releasing each one as it was completed.

Having heard this plan from Apollonius himself, who visited Pergamon, Eudemus insisted Apollonius send him each book before release. At this stage Apollonius was likely still a young geometer, who according to Pappus stayed at Alexandria with the students of Euclid (long after Euclid's time), perhaps the final stage of his education. Eudemus may have been a mentor from Appolonius' time in Pergamon.

There is a gap between the first and second prefaces. Apollonius has sent his son, also named Apollonius, to deliver the second. He speaks with more confidence, suggesting that Eudemus use the book in special study groups. Apollonius mentions meeting Philonides of Laodicea, a geometer whom he introduced to Eudemus in Ephesus, and who became Eudemus' student. Philonides 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 cosmopolitan and scholars in this "golden age of mathematics" sought employment internationally, visited each other, read each other's works and made suggestions, recommended students, and communicated via some sort of postal service. Surviving letters are abundant.

The preface to Book III is missing, and during the interval Eudemus died, says Apollonius in the preface to Book IV. Prefaces to Books IV–VII are more formal, mere summaries omitting personal information. All four are addressed to a mysterious Attalus, a choice made, Apollonius says, "because of your earnest desire to possess my works". Presumably Attalus was important to be sent Apollonius' manuscripts. One theory is that Attalus is Attalus II Philadelphus (220–138 BC), general and defender of Pergamon whose brother Eumenes II was king, and who became co-regent after his brother's illness in 160 BC and acceded to the throne in 158 BC. Both brothers were patrons of the arts, expanding the library into international magnificence. Attalus was a contemporary of Philonides and Apollonius' motive is consonant with Attalus' book-collecting initiative.

In Preface VII Apollonius 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, and Apollonius might have died before finishing it. Pappus of Alexandria, however, provided lemmas for it, so it must have been in circulation in some form.

Conics
Apollonius was a prolific geometer, turning out a large number of works. Only one survives, Conics. Of its eight books, only the first four persist as untranslated original texts of Apollonius. Books 5-7 are only available in an Arabic translation by Thābit ibn Qurra commissioned by the Banū Mūsā; the original Greek is lost. The status of Book 8 is unknown. A first draft existed, but whether the final draft was ever produced is not known. A "reconstruction" of it by Edmond Halley exists in Latin, but there is no way to know how much of it, if any, is verisimilar to Apollonius.

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.

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 , Greek ) is an imaginary "cutting"  of a cone by a plane.
 Proposition I.3: "If a cone is cut by a plane through the vertex, the section is a triangle." In the case of a double cone, the section is two triangles such that the angles at the vertex are vertical angles.
 Proposition I.4 asserts that sections of a cone parallel to the base are circles with centers on the axis.
 Proposition I.13 defines the ellipse, which is conceived as the cutting of a single cone by a plane inclined to the plane of the base and intersecting the latter in a line perpendicular to the diameter extended of the base outside the cone (not shown). The angle of the inclined plane must be greater than zero, or the section would be a circle. It must be less than the corresponding base angle of the axial triangle, at which the figure becomes a parabola.
 Proposition I.11 defines a parabola. Its plane is parallel to a side in the conic surface of the axial triangle.
 Proposition I.12 defines a hyperbola. Its plane is parallel to the axis. It cut both cones of the pair, thus acquiring two distinct branches (only one is shown).

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  which is one modern form of the equation for a parabola. The rectangle has sides  and . It was he who accordingly named the figure, parabola, "application".

The "no applicability" case is further divided into two possibilities. Given a function, , such that, in the applicability case, , in the no applicability case, either  or . In the former,  falls short of  by a quantity termed the , "deficit".  In the latter,  overshoots by a quantity termed the , "surfeit".

Applicability could be achieved by adding the deficit,  or subtracting the surfeit,  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

where  is the deficit, while an equation for the hyperbola,

becomes

where  is the surfeit.

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.

Early printed editions

The early printed editions began for the most part in the 16th century. At that time in Europe, scholarly books were expected to be in Neo-Latin. As few mathematical manuscripts were written in Latin, the editors of the early printed works translated from Greek or Arabic. Often Greek and Latin were juxtaposed, with the Greek text representing either the original or an editor's restoration. Critical commentary of the time was typically in Latin. (Earlier commentaries had been written in ancient or medieval Greek or Arabic.) Only in the 18th and 19th centuries did editions in modern languages begin to appear. A representative list of early printed editions is given below.

  A presentation of the first four books of Conics in Greek by Fredericus Commandinus with his own translation into Latin and the commentaries of Pappus of Alexandria, Eutocius of Ascalon and Serenus of Antinouplis.
  Translation by Barrow from ancient Greek to Neo-Latin of the first four books of Conics. The copy linked here, located in the Boston Public Library, once belonged to John Adams.
  A presentation of two lost but reconstructed works of Apollonius. De Sectione Rationis comes from an unpublished manuscript in Arabic in the Bodleian Library at Oxford originally partially translated by Edward Bernard but interrupted by his death. It was given to Edmond Halley, professor, astronomer, mathematician and explorer, after whom Halley's Comet later was named. Unable to decipher the corrupted text, he abandoned it. Subsequently, David Gregory (mathematician) restored the Arabic for Henry Aldrich, who gave it again to Halley. Learning Arabic, Halley created De Sectione Rationis and as an added emolument for the reader created a Neo-Latin translation of a version of De Sectione Spatii reconstructed from Pappus Commentary on it. The two Neo-Latin works and Pappus' ancient Greek commentary were bound together in the single volume of 1706. The author of the Arabic manuscript is not known. Based on a statement that it was written under the "auspices" of Al-Ma'mun, Latin Almamon, astronomer and Caliph of Baghdad in 825, Halley dates it to 820 in his "Praefatio ad Lectorem."
  Encouraged by the success of his translation of David Gregory's emended Arabic text of de Sectione rationis, published in 1706, Halley went on to restore and translate into Latin Apollonius' entire elementa conica. Books I-IV had never been lost. They appear with the Greek in one column and Halley's Latin in a parallel column. Books V-VI came from a windfall discovery of a previously unappreciated translation from Greek to Arabic that had been purchased by the antiquarian scholar Jacobus Golius in Aleppo in 1626. On his death in 1696 it passed by a chain of purchases and bequests to the Bodleian Library (originally as MS Marsh 607, dated 1070). The translation, dated much earlier, comes from the branch of Almamon's school entitled the Banū Mūsā, "sons of Musa," a group of three brothers, who lived in the 9th century. The translation was performed by writers working for them. In Halley's work, only the Latin translation of Books V-VII is given. This is its first printed publication. Book VIII was lost before the scholars of Almamon could take a hand at preserving it. Halley's concoction, based on expectations developed in Book VII, and the lemmas of Pappus, is given in Latin. The commentary of Eutocius, the lemmas of Pappus, and two related treatises by Serenus are included as a guide to the interpretation of the Conics.

Commentary and Influence

The difficulty of Conics made an intellectual niche for later commentators, each presenting 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.

Modern English commentary

Limited material about Conics was ever written in English, because European mathematicians of the 16th–18th century, including English mathematicians such as Edmund Halley and Isaac Newton, preferred Neo-Latin. In later centuries, geometry was re-established using coordinates (analytic geometry) and synthetic methods fell out of favor, so Conics direct influence on mathematical research declined.

Presentations written entirely in native English begin in the late 19th century.

Thomas Heath

Particularly influential is Thomas Heath's translation 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', parenthetically cross-referenced.

St. John's College

Heath was active in the late 19th and early 20th century, passing away in 1940, but meanwhile another point of view developed. 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 (a translation of Book IV of the Conics by Michael N. Fried was produced in 2002). 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.

Loeb Classical Library

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.

Other works

The rest of Apollonius's works are fragmentary or lost. 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.

Edmond Halley 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.

Lost and reconstructed works described by Pappus
Besides Conics, Pappus mentions other treatises of Apollonius:

 Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio")
 Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area")
 Διωρισμένη τομή, De Sectione Determinata ("Determinate Section")
 Ἐπαφαί, De Tactionibus ("Tangencies")
 Νεύσεις, De Inclinationibus ("Inclinations")
 Τόποι ἐπίπεδοι, De Locis Planis ("Plane Loci").

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:
 Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola
 Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus)
 A comparison of the dodecahedron and the icosahedron inscribed in the same sphere
 Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements
 Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of  than those of Archimedes, who calculated  as the upper limit and  as the lower limit
 an arithmetical work (see Pappus) on a system both for expressing large numbers in language more everyday than that of Archimedes' The Sand Reckoner and for multiplying these large numbers
 a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm. on Eucl. x., preserved in Arabic and published by Woepke, 1856).

Attributed ideas

Contribution to astronomy
The equivalence of two descriptions of planet motions, one using eccentrics and another deferent and epicycles, is attributed to Apollonius. Ptolemy describes this equivalence in the Almagest.

Methods of Apollonius
According to Heath, the "Methods of Apollonius" were not personal to him; whatever influence he had on later theorists was the influence 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.

When referring to golden age geometers, modern scholars use the term "method" to mean the visual, reconstructive way in which the geometer produces a result equivalent to that produced by algebra today. As a simple example, the algebraic method to compute the area of a square is to square its side length; the analogous geometric method 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 had been defined by Henry Burchard Fine in 1890 or before, who applied 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....

Nevertheless, according to Boyer, Apollonius' treatment of curves is in some ways similar to the modern treatment, and his work seems to anticipate analytical geometry.
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 Leibniz in Acta Eruditorum, 1684. Here is an example from Conics, Book I, on Proposition 11:

Literal translation of the Greek: Let it be contrived that the (square) of BC be to the (rectangle) of BAC as FH is to FA
Taliaferro's translation: "Let it be contrived that sq. BC : rect. BA.AC :: FH : FA"
Algebraic equivalent: BC2/BA•BC = FH/FA

References

Notes

Citations

Sources

Apollonius de Perge, La section des droites selon des rapports, Commentaire historique et mathématique, édition et traduction du texte arabe. Roshdi Rashed and Hélène Bellosta, Scientia Graeco-Arabica, vol. 2. Berlin: De Gruyter, 2010.

Further reading

External links

 Scans of old editions of some of Apollonius' works in several languages at wilbourhall.org

Ancient Greek astronomers
Ancient Greek geometers
Ancient Greek inventors
Pamphylians
History of geometry
260s BC births
190s BC deaths
3rd-century BC mathematicians
2nd-century BC mathematicians

Apollonius of Perga


Arnold Odermatt (25 May 1925 − 19 June 2021) was a Swiss police photographer whose work spanned more than 40 years. Originally trained as a baker, he was a photographer for the Nidwalden cantonal police from 1948 until his retirement in 1990. He is best known for his eerily beautiful black and white photographs of the aftermaths of motor vehicle accidents. Odermatt joined the police in 1948 and rose to become a lieutenant, chief of the transport police, and deputy chief inspector of the Nidwalden Police before he retired.

At the beginning of the 1990s, Odermatt's photography was discovered by his son, Urs Odermatt during research for his film , and this work became a central theme in the film's plot. Urs brought his father's works together in the working groups entitled Meine Welt, Karambolage, Im Dienst, and In zivil and has published Odermatt's work ever since, working in collaboration with the Frankfurt art historian Beate Kemfert and a gallery in Berlin—Galerie Springer & Winckler.

In 2001, Odermatt's photography was selected by Harald Szeemann to be exhibited at the 49th Venice Biennale. In 2002 James Rondeau exhibited Odermatt's work in its own right at the Art Institute of Chicago, as did Urs Stahel at the Fotomuseum Winterthur in 2004.

Biography
Odermatt was born in Oberdorf, canton Nidwalden, Switzerland. He joined the Nidwalden Police in 1948. He was forced to give up his original career as a bakery and pastry chef on health grounds. As the policeman Odermatt first appeared with his Rolleiflex at the scene of an accident – to provide photos to complement the police report, people found this rather disconcerting. At that time, photography was anything other than an independent means of providing the police with evidence.

A colleague observed Odermatt as he took pictures for the force and was suspicious. He was ordered to report to his commander immediately. Odermatt managed to convince his superiors of the pioneering work he was doing. They allowed him to convert an old toilet in an observation post in Stans into a makeshift darkroom. When the observation post was moved into another building several years later, Switzerland's first police photographer was given his own laboratory.

Odermatt's biggest role model was the famous Magnum photographer Werner Bischof. He met him once by chance, as he was on security duty on the Bürgenstock and wanted to photograph Charlie Chaplin. Odermatt's own style was characterised by sobriety and authenticity. The spartan linguistic expression of his police reports can also be found in Odermatt's images. His craftsmanship is beyond question, nothing of note is missed by his photographic eye. In KARAMBOLAGE, his most famous series of work, you can't see the maimed victims but you do see the ethereal, surreal sculptures of scrap metal. With the softness and melancholy of Jacques Tati, he looks at the consequences of speed and the hectic nature of modern times.

For 40 years, Odermatt captured the daily work of the Nidwalden police force. It was only rarely that the local press, the court or an insurance company were interested in his photos. It was only when his son, the film and theatre director Urs Odermatt, showed the photos  for the first time at a solo exhibition in Frankfurt am Main that the art scene first became interested in his work. After the inspiring exhibition, the photo book 'Meine Welt' followed. Suddenly the everyday observations from the central Swiss province had gained the same status as those of his well-travelled predecessor, Werner Bischof.

At an early stage in his police career, when Arnold used the camera to catalogue traffic accidents, this was a revolutionary innovation in the Swiss police. If Odermatt were to turn up at a crime scene with his camera today, he could expect to be told that photography was not for him, but was instead the job of a specially trained police photographer.

Solo exhibitions
 1993 - Arnold Odermatt. Seeplatz 10, Buochs.
 1996 - Meine Welt. Viewpoint Gallery, Salford.
 1998 - Karambolage. Police headquarters, Frankfurt am Main.
 2000 - Karambolagen und andere Photographien. Springer & Winckler Galerie, Berlin.
 2001 - Carambolages. Centre de la photographie, Genf.
 2002 - Karambolage. Museum Morsbroich, Leverkusen. .
 2002 - Die Biennale-Auswahl. 32 Photographien für Venedig 2001. Springer & Winckler Galerie, Berlin. .
 2002 - Karambolage. Centre rhénan de la photographie, Strassburg.
 2002 - Arnold Odermatt. The Art Institute of Chicago, Chicago. Catalogue.
 2003 - Arnold Odermatt. Washington University Gallery of Art, St. Louis. Katalog.
 2003 - Arnold Odermatt. Paul Morris Gallery, New York.
 2004 - Arnold Odermatt. James Kelly Contemporary, Santa Fe.
 2004 - Arnold Odermatt. Galerie Sabine Knust, Munich.
 2004 - Karambolagen. Photo museum Winterthur, Winterthur.
 2004 - Karambolage. Rathaus, Fellbach.
 2004 - Arnold Odermatt. Galería Arnés y Röpke, Madrid.
 2004 - Kantonspolizei Nidwalden. Springer & Winckler Galerie, Berlin.
 2005 - Arnold Odermatt. Centro Cultural Okendo, San Sebastián. .
 2005 - Arnold Odermatt. Museum im Bellpark, Kriens.
 2006 - Im Dienst. Farbphotographien 1962-1990. Springer & Winckler Galerie, Berlin.
 2008 - Focus Photographie: Arnold Odermatt. Galerie Lelong, Zurich.
 2009 - Project room: Arnold Odermatt. Galerie Georges-Philippe & Nathalie Vallois, Paris.
 2009 - Karambolage. Miesiąc Fotografii w Krakowie, Krakau. .
 2010 - Arnold Odermatt. Leo Koenig Inc. Projects, New York.
 2010 - In zivil. Springer & Winckler Galerie, Berlin.
 2010 - On Duty. Amador Gallery, New York.
 2011 - On and Off Duty. Galerie Georges-Philippe & Nathalie Vallois, Paris.
 2011 - Heimat. Buchmann Galerie, Berlin.
 2012 - Arnold Odermatt. L'Été photographique de Lectoure. Lectoure, (Gascogne).
 2012 - Arnold Odermatt. Espace d'Art Contemporain Fernet Branca, Saint-Louis (Elsass).
 2012 - Arnold Odermatt. La Chambre, Straßburg.

Books
 Urs Odermatt: Wachtmeister Zumbühl. Script for a feature film with 79 stills photographs by Odermatt. Benteli Verlag, Bern 1994. .
 Arnold Odermatt: Karambolage. Museum Morsbroich, Leverkusen 2002. .
 Arnold Odermatt: Die Biennale-Auswahl. 32 Photographien für Venedig 2001. [Biennale Selection: 32 photographs for Venice 2001] With a text by Harald Szeemann. Galerie Springer & Winckler, Berlin 2002. .
 Arnold Odermatt: Meine Welt. Photographien/Photographs 1939-1993. Edited by Urs Odermatt. Benteli Verlag, Bern 1993, 2001 und 2006. . Kodak-Fotobuch-Preis 1993.
 Arnold Odermatt: Karambolage. Edited by Urs Odermatt. German, French English. Steidl-Verlag, Göttingen 2003. .
 Arnold Odermatt: Im Dienst. En service. On Duty. Edited by Urs Odermatt. Steidl Verlag, Göttingen 2006. . German Photobook Prize 2008.
 Arnold Odermatt: In zivil. Hors service. Off Duty. Edited by Urs Odermatt. Steidl-Verlag, Göttingen 2010. .
 Arnold Odermatt. With a text by Caroline Recher. Diaphane éditions, Montreuil sur Brèche 2012, .
 Arnold Odermatt: Karambolage. Steidl Verlag. Göttingen 2013,

Films
In the 1960s Odermatt documented the early construction of the Swiss motorways in Acheregg and the Lopper tunnel, with extensive photos and 16mm black-and-white film footage. In 1991 Urs Odermatt put all of this historical film material together in the documentary film Lopper.

In the 1990s, working as a stills photographer, during filming, Odermatt worked on the feature films Rotlicht!, Gekauftes Glück und Wachtmeister Zumbühl by Urs Odermatt.

References

External links
Official website Arnold Odermatt

Artnet database: Arnold Odermatt
Steidl Verlag: Arnold Odermatt - Karambolage
Sammlung Fotomuseum Winterthur collection - Arnold Odermatt
Edition 5: art postcards by Arnold Odermatt
Lopper, documentary film. Camera: Arnold Odermatt
WDR: Bei Odermatt wird die Karambolage zur Kunst
TAZ: Die Stille nach dem Crash!
WOZ: Arnold Odermatt und John Waters im Fotomuseum Winterthur
news.ch: Trash und Crash im Fotomuseum Winterthur - Videobeitrag & Interview mit Arnold Odermatt

1925 births
People from Nidwalden
20th-century Swiss photographers
2021 deaths
Swiss police officers
20th-century Swiss male artists
21st-century Swiss male artists

Arnold Odermatt


A portmanteau is a piece of luggage, usually made of leather and opening into two equal parts. Some are large, upright, and hinged at the back and enable hanging up clothes in one half, while others are much smaller bags (such as Gladstone bags) with two equally sized compartments.

The word derives from the French word portemanteau (from porter, "to carry", and manteau, "coat") which nowadays means a coat rack but was in the past also used to refer to a traveling case or bag for clothes.

Portmanteau mail bag

In the 1700s the term would also have described a Mail bag. This continued in the 1800s for bags used by the Post Office. An 1823  resolution in Congress further stated that "locks...will be placed on the portmanteaus containing the principal mails (which) can only be opened...at the distributing offices."

References

External links

Luggage

Portmanteau (luggage)


Toyaguda, formerly called Kora, is a small village in Bela tehsil, Adilabad district, Telangana state, India. It is located about 330 km north of Hyderabad, 24 km east of Adilabad, and 15 km southwest of Bela, and 1 km west of the Sathnala Dam Reservoir.

According to the 2011 census, the town had 268 households and 1296 inhabitants (including 113 children 0–6), with an adult literacy rate of 66%.

References

Villages in Adilabad district

Toyaguda, Adilabad


The Church of the Advent is a historic church building on Church Street, one block south of the junction with Maine State Route 229 in Limestone, Maine.  Built in 1881, the architecturally eclectic Gothic/Romanesque building  was constructed for an Episcopalian church congregation founded in 1872, which is believed to be the oldest in Limestone.  The church was listed on the National Register of Historic Places in 1991.

Architecture and building history
The Church of the Advent is a modest wood frame building, with a distinctive steeply gabled roof and a small tower just to the left of the main block.  The main (west-facing) facade has two tall and narrow round-headed windows, with the main entrance in a rounded opening in the tower.  A narrow rectangular window is above the doorway in the tower, with an open belfry above, capped by a pyramidal roof.  Behind the sanctuary is a wider modern single-story addition built of concrete blocks, with a shallow-pitch gable roof.  The interior (walls and barrel-vaulted ceiling) are finished in stained tongue-and-groove woodwork, although early accounts describe the church ceiling as having been plastered.  Surviving original features include wall-mounted oil lamps.

The Episcopal congregation that built the church in 1881 was founded in 1872, and is believed to be Limestone's first organized church group.  It originally met in a school before having this building built.  The building was originally located on the Grand Falls Road where the Protestant Cemetery continues to stand. The church was moved on log rollers, using horses to its present location over several months in 1896.  It was reported in 1904 that its tower had been raised to accommodate the belfry.

See also
National Register of Historic Places listings in Aroostook County, Maine

References

External links
Aroostook Cluster of Episcopal Churches web site

Churches in Aroostook County, Maine
Churches on the National Register of Historic Places in Maine
Gothic Revival church buildings in Maine
Churches completed in 1881
19th-century Episcopal church buildings
Episcopal church buildings in Maine
National Register of Historic Places in Aroostook County, Maine
1881 establishments in Maine

Church of the Advent (Limestone, Maine)


Allen Ezail Iverson (/ˈaɪvərsən/; born June 7, 1975),  is an former American professional basketball player. He played for 14 seasons in the National Basketball Association (NBA) at both the shooting guard and point guard positions. Iverson was an 11-time NBA All-Star, won the All-Star game MVP award in 2001 and 2005, and was the NBA's Most Valuable Player (MVP) of the 2000–01 season. He was inducted into the Naismith Memorial Basketball Hall of Fame in 2016.

Education
Iverson went to Bethel High School in Hampton, Virginia. This is where he played for the football team and the basketball team. He won the Associated Press High School Player of the Year award in both football and basketball. He also won the Division AAA Virginia state championship in both sports. After high school, Iverson won a scholarship to Georgetown University. He played college basketball with the Georgetown Hoyas for two years. He set the school record for career scoring average (22.9 points per game). He also won the Big East Defensive Player of the Year awards in both years.

Career
After two years at Georgetown, Iverson was eligible for the 1996 NBA draft. He was selected by the Philadelphia 76ers with the first overall pick. He was named the NBA Rookie of the Year in the 1996–97 season. Iverson won the NBA scoring title during the 1998–99, 2000–01, 2001–02, and 2004–05 seasons. Iverson was one of the smallest players in NBA history at 6 feet, 0 inches. His regular season career scoring average of 26.7 points per game ranks seventh all-time, and his playoff career scoring average of 29.7 points per game is second after Michael Jordan. Iverson led his team to the 2001 NBA Finals. He won the bronze medal at the 2004 Summer Olympics.

Later in his career, Iverson played for the Denver Nuggets, Detroit Pistons, and the Memphis Grizzlies, before ending his NBA career with the 76ers during the 2009–10 season. He was ranked as the fifth greatest NBA shooting guard of all time by ESPN in 2008. He finished his career playing with Turkish-based basketball team, Beşiktaş in 2011. In 2017, Iverson joined BIG3, a 3-on-3 basketball league, as a player-coach for the team '3's Company.'

1975 births
Living people
American basketball players
African-American basketball players
Sportspeople from Virginia
Naismith Memorial Basketball Hall of Fame inductees

Allen Iverson


Grossmont College is a public community college in El Cajon, California. Its name originated with the silent film actor and producer William J. Gross, who was enticed by Ed Fletcher to invest in the purchase of land, part of which was called Grossmont. The campus sits in the Fletcher Hills community of El Cajon and is bordered by the cities of San Diego and Santee. Grossmont College along with Cuyamaca College make up what is the Grossmont-Cuyamaca Community College District. Grossmont is part of the California Community College System.

Grossmont College is also home to Grossmont Middle College High School, where selected high school students can receive both high school and college credit for taking courses on campus. The newspaper for Grossmont College is The Summit. Its radio station is Griffin Radio.

Facilities

The facilities of Grossmont College are situated across 135 acres. At its inception, the campus was planned to accommodate an enrollment of 2,500 daytime students. The first incarnation of the completed campus was expected to hold 4,800 students. On October 18, 1965, a bond for $3.5 million was approved by area voters. This made it possible to complete the college's initial master plan. By September 25, 1967, the new facilities were completed. Since that time, student enrollment increased dramatically and created the need for new and remodeled campus facilities.

In recent years, the college has undergone major improvements of its facilities including:

Academics
Grossmont College offers more than 150 degree and certificate programs. Each year, about 1,500 students earn Associate in Arts degrees, Associate in Science degrees, and/or advanced/basic certificates. In addition, the students are offered general education and transfer programs.

Athletics

Grossmont College offers several men's and women's intercollegiate sports:

Notable alumni
 Sergio Bailey II - NFL and XFL football player
Lester Bangs - Music journalist
 Doug Benson - Comedian
 Quintin Berry - MLB Player, Boston Red Sox
 Rachel Bilson - American actress
 Brian Patrick Butler - Actor and filmmaker
 Brad Daluiso - American football player
 Chad DeGrenier - American football player and coach
 Stephanie Nicole Garcia-Colace - Professional wrestler, actress, known as Nikki Bella
Mark Goffeney (1969-2021) - Armless guitar player, child celebrity, disability advocate
 Robert Hays - American actor
 Arthur Hobbs - Defensive back for the Hamilton Tiger-Cats
 Barry Jantz - Former La Mesa City Councilman/CEO of Grossmont Healthcare District
Brian Jones (born 1968) - politician serving in the California State Senate
 David Leisure - Actor
 Kevin McCadam - NFL defensive back, Atlanta Falcons
 Dan Melville - American football player
 Sean O'Sullivan - MLB Player, Boston Red Sox
 Dat Phan - Vietnamese American stand-up comedian
 Joe Roth - College football quarterback
 Wally Schlotter - Chairman of the San Diego Film Commission from 1978 to 1996
 Cathy Scott - True crime author and journalist
 Bernard Seigal - Musician, Beat Farmers' original member, music journalist, editor college newspaper The Summit
 Scott Sherman - former San Diego City Council member
 Brian Sipe - NFL quarterback, Cleveland Browns
 Alexandra Slade - Actress
 Akili Smith - NFL quarterback, Cincinnati Bengals
 Casey Tiumalu - NFL running back, Los Angeles Rams
 Todd Watkins - NFL wide receiver

References

External links

California Community Colleges
Education in El Cajon, California
Universities and colleges in San Diego County, California
Schools accredited by the Western Association of Schools and Colleges
1961 establishments in California
Universities and colleges established in 1961
Two-year colleges in the United States

Grossmont College


is a Japanese game designer and producer at Nintendo. He created the Donkey Kong, Mario, Star Fox, Pikmin and The Legend of Zelda video game series for Nintendo.

He is one of the most famous game designers in the world and is often called the father of modern video gaming. His games give players many ways to play and explore. This was unique in video games when his games were first released.

Miyamoto started working with Nintendo in 1977 as an artist when it was still a toy and playing-card company. In 1980, he designed Donkey Kong, which was a big success. Miyamoto became Nintendo's leading producer, and he continues to create many popular games for Nintendo today Such as entries in the Super Mario and the Zelda series.

Early Life

Miyamoto was born on November 16, 1952 in the Japanese town of Sonobe, which is a rural town that is northwest of Kyoto. His father taught English language.

Miyamoto graduated from Kanazawa Municipal College of Industrial Arts with a degree in industrial design. However, he could not find a job. He also loved manga. Before he thought about starting a career in video games, he wanted to become a professional manga artist. Manga's classical kishōtenketsu narrative structure influenced him. Western genre television shows also influenced him. The game that made him want to enter the video game industry was the 1978 arcade game Space Invaders.

Career

1977–1984: Arcade beginnings and Donkey Kong

Nintendo was a small Japanese company in the late 1970s. They had traditionally sold playing cards and other novelties. However, they started making toys and games in the mid-1960s. Through a mutual friend, Miyamoto's father set up an interview with Nintendo president Hiroshi Yamauchi. After showing some of his toy creations, Miyamoto was hired in 1977 as an apprentice in the planning department.

Miyamoto became Nintendo's first artist. He helped create the art for Nintendo's first original coin-operated arcade video game, Sheriff. He first helped the company develop a game with the 1980 release Radar Scope. The game had some success in Japan. However, by 1981, Nintendo's efforts to sell it in the North American video game market failed. This left the company with a lot of arcade machines that were not sold. Nintendo was also facing financial ruin. In order to try to keep Nintendo going, Nintendo president Hiroshi Yamauchi decided to change unsold Radar Scope machines into a new arcade game. He asked Miyamoto to do this. Miyamoto said self-deprecatingly that "no one else was available" to do the work. Nintendo's head engineer, Gunpei Yokoi, oversaw the project.

Miyamoto thought of many characters and story ideas, but eventually decided on a love triangle between a gorilla, a carpenter, and a girl. He wanted to create something like what was with comic characters Bluto and Popeye for the woman Olive Oyl. Although, Nintendo tried to get the rights to Popeye, but they failed. Bluto became an ape, a form Miyamoto said was "nothing too evil or disgusting". This ape would be the pet of the main character, "a funny, hang-loose kind of guy." Miyamoto also said "Beauty and the Beast" and the 1933 film King Kong influenced him. Donkey Kong was the first time that the creation of a video game's story came before the actual programming. Usually, the story was added without much care. Miyamoto had high hopes for his new project. However, he did not actually know how to program it himself; instead, he thought of the game's concepts, then asked technicians on whether they were possible. He wanted to make the characters different sizes, move in different ways, and react in various ways. However, Yokoi thought Miyamoto's original design was too difficult. Yokoi suggested using see-saws to catapult the character across the screen; however, this was also too difficult to do. Miyamoto next thought of using sloped platforms and ladders to move the character. He thought of using barrels for obstacles. When he asked that the game have multiple stages, the four-man programming team complained that he was basically asking them to make the game repeat, but the team eventually programmed the game. When the game was sent to Nintendo of America for testing, the sales manager did not like how it was different from the maze and shooter games common at the time. When American staffers started naming the characters, they decided on "Pauline" for the woman. She was named after Polly James, wife of Nintendo's Redmond, Washington, warehouse manager, Don James. The playable character was first called "Jumpman". They changed the name to Mario. He was named after Mario Segale, the warehouse landlord. These character names were written on the American cabinet art and used in advertising. The staff also got an English name for the game. The title was Donkey Kong.

Donkey Kong was successful. Miyamoto started to work on sequels Donkey Kong Jr. in 1982 and Donkey Kong 3 in 1983. In the next game, he changed the Donkey Kong character Jumpman into Mario. He gave him a brother: Luigi. He named the new game Mario Bros. Yokoi convinced Miyamoto to give Mario some special powers, namely being able to fall from any height without getting hurt. Mario's look in Donkey Kong—overalls, a hat, and a thick mustache—led Miyamoto to change parts of the game. He changed it to make Mario look like a plumber instead of a carpenter. Miyamoto thought that New York City would be the best setting for the game. He thought it was the best place because it has a "maze of subterranean sewage pipes". The two-player mode and other parts of gameplay were influenced by an earlier video game called Joust. To date, games in the Mario Bros. series have been released for many consoles. Shortly after, Miyamoto also drew the character sprites and game design for the Baseball, Tennis, and Golf games on the NES.

1985–1989: NES/Famicom, Super Mario Bros., and The Legend of Zelda

Nintendo released its first home video game console, the Family Computer (released in North America as the Nintendo Entertainment System). While this happened, Miyamoto created two of the most famous games for the console and in the history of video games as a whole: Super Mario Bros. (a sequel to Mario Bros.) and The Legend of Zelda (an entirely original title).

In both games, Miyamoto decided to focus more on gameplay than on high scores. This was different from most games at the time. Super Mario Bros. was a linear game. The player moved the stage by running, jumping, and dodging or defeating enemies. Miyamoto used nonlinear gameplay in The Legend of Zelda. This requires the player to think their way through riddles and puzzles. The world was big and it seemed to never end. It had a lot of different choices and depth never seen before in a video game. With The Legend of Zelda, Miyamoto wanted to make an in-game world that players could relate to. He said it was like a "small garden that they can put inside their drawer." He was influenced from his time as a boy around Kyoto. When he was a boy, he explored nearby fields, woods, and caves; each Zelda title has this feeling of exploration. "When I was a child," Miyamoto said, "I went hiking and found a lake. It was quite a surprise for me to find it. When I traveled around the country without a map, trying to find my way, finding amazing things as I went, I realized how it felt to go on an adventure like this." He recreated his memories of becoming lost in the maze of sliding doors in his family home in Zeldas maze-like dungeons. In February 1986, Nintendo released the game as the first game for the Nintendo Entertainment System's new Disk System.

Miyamoto worked on different games for the Nintendo Entertainment System. These games included Ice Climber, Kid Icarus, Excitebike, and Devil World. He also worked on sequels to Super Mario Bros and The Legend of Zelda.

Super Mario Bros. 3 was developed by Nintendo Entertainment Analysis & Development; the game took more than two years to make.

1990–2000: SNES, Nintendo 64, Super Mario 64, and Ocarina of Time

A merger between Nintendo's different internal research and development teams led to the creation of Nintendo Entertainment Analysis & Development (Nintendo EAD). Miyamoto was the head of it. Nintendo EAD had about fifteen months to develop F-Zero, one of the launch titles for the Super Nintendo Entertainment System. Miyamoto worked on different games on the Super Nintendo Entertainment System. One of them was Star Fox. For the game, programmer Jez San convinced Nintendo to make an upgrade for the Super Nintendo. This upgrade would allow it to be able to show three-dimensional graphics better. This upgrade was the Super FX chip. Using this new hardware, Miyamoto and Katsuya Eguchi designed the Star Fox game with an early use of three-dimensional graphics.

Miyamoto produced two big Mario titles for the system. The first game was Super Mario World. It was a launch title. It was included with Super Nintendo Entertainment System consoles. It had an overworld as in Super Mario Bros.. It had a new character, Yoshi. Yoshi would go on to be in many other Nintendo games. The second Mario game for the system was Super Mario RPG. It was different than most Mario games. Miyamoto led a team that was a partnership between Nintendo and Square Co.; it took almost a year to develop the graphics. The story takes place in a newly rendered Mushroom Kingdom based on the Super Mario Bros. series.

Shigeru Miyamoto helped Satoshi Tajiri. Miyamoto guided him during the creation of Pocket Monsters: Red and Green (released in English as Pokémon Red and Blue), the first video games in the Pokémon series. He was also the producer for these games. Miyamoto worked on social gameplay ideas such as trading. Pokémon would become one of the most popular entertainment franchises in the world, spanning video games, anime, and other merchandise.

Miyamoto made several games for the Nintendo 64. Most of these games were from his previous franchises. His first game on the Nintendo 64 was Super Mario 64. He was the main director. In developing the game, he started with character design and the camera system. Miyamoto and the other designers were unsure of what the game should be like. It took months to decide on a good camera view and layout. The first idea used a fixed path. This would have been similar to an isometric-type game.They decided on a free-roaming 3D design. He changed the design of the Nintendo 64 controller to fit with that of Super Mario 64.

Using what he had learned about the Nintendo 64 from developing Super Mario 64 and Star Fox 64, Miyamoto produced his next game, The Legend of Zelda: Ocarina of Time. He led a team of several directors. Individual parts of Ocarina of Time were developed by multiple directors—a new strategy for Nintendo EAD. However, when things progressed slower than expected, Miyamoto returned to the development team with a more central role. He was helped in public by interpreter Bill Trinen. The team was new to 3D games. However, assistant director Makoto Miyanaga remembers a feeling of "passion for creating something new and never done before". Miyamoto went on to produce a sequel to Ocarina of Time, known as The Legend of Zelda: Majora's Mask.

Miyamoto worked many Mario series spin-offs for the Nintendo 64. These included Mario Kart 64 and Mario Party.

2000–2011: GameCube, Wii, and DS

Miyamoto produced many games for the GameCube. This included the launch title Luigi's Mansion. The game was first shown at Nintendo Space World 2000 as a technical demo. It was designed to show off the graphical power of the GameCube. Miyamoto made a short demo of the game concepts. Nintendo decided to make it a full game. Luigi's Mansion was later shown at E3 2001 with the GameCube console. Miyamoto made more Mario spinoffs in these years. He also produced the 3D game series Metroid Prime. This was because the original designer Yokoi, a friend of Miyamoto's, died. In this time he developed Pikmin and its sequel Pikmin 2. These games were based on his experiences gardening. He also worked on new games for the Star Fox, Donkey Kong, F-Zero, and The Legend of Zelda series on the GameCube and the Game Boy Advance systems. With the help of Hideo Kojima, he helped the developers of Metal Gear Solid: The Twin Snakes. He helped with many games on the Nintendo DS. The games included the remake of Super Mario 64, Super Mario 64 DS. It also included the new game Nintendogs, a new franchise based on his own experiences with dogs.

2011–present: Wii U, 3DS, and Switch
Miyamoto produced Super Mario 3D Land and Luigi's Mansion: Dark Moon for the 3DS, and Pikmin 3 for the Wii U.

After Nintendo president Satoru Iwata died in July 2015, Miyamoto was chosen to be an acting Representative Director, along with Genyo Takeda. He was no longer in this position in September 2015 when Tatsumi Kimishima became the company's president. He was also chosen to be in the position of "Creative Fellow" at the same time. He would give expert advice to Kimishima as a "support network" alongside Takeda.

Miyamoto was a creative director on the 2017 game Super Mario Odyssey. He was not one of its producers. He is credited as being a big influence on the game's development.

Development philosophy
Miyamoto and Nintendo do not use focus groups. Instead, Miyamoto figures out if a game is fun by playing it himself. He says that if he enjoys it, others will too. He uses how the Pokémon series was created as an example. He said "And that's the point – Not to make something sell, something very popular, but to love something, and make something that we creators can love. It's the most important feeling we should have when we make games."

Impact

Time called Miyamoto "the Spielberg of video games" and "the father of modern video games." The Daily Telegraph says he is "regarded by many as possibly the most important game designer of all time." GameTrailers called him "the most influential game creator in history." Miyamoto has greatly influenced various things about video games in general. The Daily Telegraph credited him with creating "some of the most innovative, ground breaking and successful work in his field." Many of Miyamoto's works have started new video game concepts or changed existing ones. Miyamoto's games have received outstanding critical praise. Some are considered to be the best games of all time.

Miyamoto's games have also sold very well. Some became the best-selling games on Nintendo consoles and of all time. As of 1999, his games had sold 250 million copies and grossed billions of dollars.

Critical reception
Miyamoto's games have received a lot of praise. Many of his games are considered some of the best of all time.

Games in Miyamoto's The Legend of Zelda series have received a lot of praise. A Link to the Past was a big game for Nintendo. It is considered today to be one of the best video games ever made. Ocarina of Time is considered by critics and gamers to be one of the best video games ever made. Twilight Princess was praised by nearly everyone who reviewed it. It is the third highest-rated game for the Wii. It received perfect scores from major publications such as CVG, Electronic Gaming Monthly, Game Informer, GamesRadar, and GameSpy.

Many critics and fans say Super Mario 64 is one of the best and most revolutionary video games of all time.

Metacritic says that Super Mario Galaxy and Super Mario Galaxy 2 are the first- and second-highest rated games for the Wii.

A 1995 article in Maximum said that "in gaming communities, Miyamoto's name is more famous than Steven Spielberg's could ever be."

Commercial reception
Miyamoto's games have sold very well. many of them became some of the best-selling games on Nintendo consoles and of all time.

Miyamoto's Mario series is the best-selling video game franchise of all time. It has sold over 400 million units. Super Mario Bros. is the second best-selling video game of all time. Super Mario Bros., Super Mario Bros. 3, and Super Mario Bros. 2 were, respectively, the three best-selling games for the Nintendo Entertainment System. Levi Buchanan of IGN considered Super Mario Bros. 3s appearance in the film The Wizard as a show-stealing part. He said the movie was a "90-minute commercial" for the game. Super Mario World was the best-selling game for the Super Nintendo Entertainment System. Super Mario 64 was the best-selling Nintendo 64 game. As of May 21, 2003, the game had sold eleven million copies. At the end of 2007, Guinness World Records said the sales were 11.8 million copies. As of September 25, 2007, it was the seventh best-selling video game in the United States because it had sold six million copies. By June 2007, Super Mario 64 had become the second most popular title on Wii's Virtual Console. Super Mario Bros. was the most popular on the Wii Virtual Console. Super Mario Sunshine was the third best-selling video game for the GameCube.

The first The Legend of Zelda game was the fifth best-selling game for the Nintendo Entertainment System. The Wind Waker was the fourth best-selling game for the GameCube. Twilight Princess had sold well. In the PAL region, which covers most of Asia, Africa, South America, Australia, New Zealand, and most of Western Europe, Twilight Princess is the best-selling Zelda game ever. During its first week, the game was sold with three out of every four Wii purchases. The game had sold 4.52 million copies on the Wii as of March 1, 2008. 1.32 million copies were sold on the GameCube as of March 31, 2007.

Awards and recognition

The name of the main character of the PC game Daikatana, Hiro Miyamoto, is for Miyamoto. The character Gary Oak from the Pokémon anime series is named Shigeru in Japan.  He is the rival of Ash Ketchum (called Satoshi in Japan). Pokémon creator Satoshi Tajiri was mentored by Miyamoto.

In 1998, Miyamoto was the first person to be in the Academy of Interactive Arts and Sciences' Hall of Fame. In 2006, Miyamoto was made a Chevalier (knight) of the French Ordre des Arts et des Lettres by the French Minister of Culture Renaud Donnedieu de Vabres.

On November 28, 2006, Miyamoto was in TIME Asia's "60 Years of Asian Heroes," alongside Hayao Miyazaki, Mahatma Gandhi, Mother Teresa, Bruce Lee and the Dalai Lama. He was later one of Time Magazine's 100 Most Influential People of the Year in both 2007 and also in 2008. At the Game Developers Choice Awards, on March 7, 2007, Miyamoto received the Lifetime Achievement Award. Both GameTrailers and IGN placed Miyamoto first on their lists for the "Top Ten Game Creators" and the "Top 100 Game Creators of All Time".

He was made a Fellow of BAFTA at the British Academy Video Games Awards on March 19, 2010. In 2012, Miyamoto was the first interactive creator to be awarded the highest recognition in the Spanish-speaking world, the Prince of Asturias Award.

Personal Life
Miyamoto has a wife, Yasuko, and two children. In 2010, his son was 25 and working at an advertising agency. His daughter was 23 and studying zoology at the time. His children played video games in their youth, but he also made them go outside. Although he knows some English, he is not fluent and would rather speak in Japanese for interviews.

Miyamoto usually does not sign autographs. He is worried that if he signs an autograph for one person, many other people will ask for autographs too. He also does not appear on Japanese television because he likes privacy. More foreign tourists than Japanese people recognize him.

Miyamoto does not spend much time playing video games in his personal time. He likes playing the guitar, mandolin, and banjo. He likes bluegrass music. He has a Shetland Sheepdog named Pikku. The dog was why Miyamoto wanted to create Nintendogs. He is also a semi-professional dog breeder. He has been quoted as saying, "Video games are bad for you? That's what they said about rock and roll."

Miyamoto like to move the furniture in his house, even late at night. He also said that he has a hobby of guessing how long something is, then checking to see if he was correct. He carries a measuring tape with him everywhere. In December 2016, Miyamoto showed his hobby on The Tonight Show Starring Jimmy Fallon. He also played the Super Mario Bros. theme on guitar with The Roots during the same show.

Games created

References

Other websites

 Shigeru Miyamoto on Nintendo Miiverse
 "Master of Play" profile in the New Yorker, December 20, 2010
 New York Times profile, May 25, 2008
  from the digital TV series Play Value
Shigeru Miyamoto at RAWG

1952 births
Living people
Japanese video game designers
People from Kyoto Prefecture
Nintendo people

Shigeru Miyamoto