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Apollonius of Perga

Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος ; c. 240 BC) 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eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.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.

"Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. And on one occasion, when looking into the tract written by Apollonius about the comparison of the dodecahedron and icosahedron inscribed in one and the same sphere, that is to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius' treatment of it in this book was not correct; accordingly, as I understood from my father, they proceeded to amend and rewrite it. But I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, and I was greatly attracted by his investigation of the problem. Now the book published by Apollonius is accessible to all; for it has a large circulation in a form which seems to have been the result of later careful elaboration."

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 Museum 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 sectio, Greek tome) 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 $y^2 = kx$ which is one modern form of the equation for a parabola. The rectangle has sides $$k$$ and $$x$$. It was he who accordingly named the figure, parabola, "application".

The "no applicability" case is further divided into two possibilities. Given a function, $f(x)$, such that, in the applicability case, $y^2 = g(x)$ , in the no applicability case, either $y^2 > g(x)$ or $y^2 < g(x)$. In the former, $$g(x)$$ falls short of $y^2$ by a quantity termed the ellipsis, "deficit". In the latter, $$g(x)$$ overshoots by a quantity termed the hyperbole, "surfeit".

Applicability could be achieved by adding the deficit, $y^2 = f(x) = g(x) + d,$ or subtracting the surfeit, $$g(x) - s.$$ 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,

 * $$Ax^2 + By^2 = C$$

can be placed in the form

 * $$y^2 = \left| \frac AB x^2 \right| + \frac CB$$

where $$C/B$$ is the deficit, while an equation for the hyperbola,

 * $$Ax^2 - By^2 = C$$

becomes

 * $$y^2 = \left| \frac AB x^2 \right| - \frac CB$$

where $$C/B$$ 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 $\pi$ than those of Archimedes, who calculated $3 1/7$ as the upper limit and $3 10/71$ 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).

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

title: Apollonius of Perga


text:
Pentaquark

A pentaquark is a human-made subatomic particle, consisting of four quarks and one antiquark bound together; they are not known to occur naturally, or exist outside of experiments specifically carried out to create them.

As quarks have a baryon number of $\overline{q}$, and antiquarks of $+ 1⁄3$, the pentaquark would have a total baryon number of 1, and thus would be a baryon. Further, because it has five quarks instead of the usual three found in regular baryons (a.k.a. "triquarks"), it is classified as an exotic baryon. The name pentaquark was coined by Claude Gignoux et al. (1987) and Harry J. Lipkin in 1987; however, the possibility of five-quark particles was identified as early as 1964 when Murray Gell-Mann first postulated the existence of quarks. Although predicted for decades, pentaquarks proved surprisingly difficult to discover and some physicists were beginning to suspect that an unknown law of nature prevented their production.

The first claim of pentaquark discovery was recorded at LEPS in Japan in 2003, and several experiments in the mid-2000s also reported discoveries of other pentaquark states. However, other researchers were not able to replicate the LEPS results, and the other pentaquark discoveries were not accepted because of poor data and statistical analysis. On 13 July 2015, the LHCb collaboration at CERN reported results consistent with pentaquark states in the decay of bottom Lambda baryons (Λ$− 1⁄3$). On 26 March 2019, the LHCb collaboration announced the discovery of a new pentaquark that had not been previously observed. On 5 July 2022, the LHCb collaboration announced the discovery of the P$0 b$(4338)0 pentaquark.

Outside of particle research laboratories, pentaquarks might be produced naturally in the processes that result in the formation of neutron stars.

Background
A quark is a type of elementary particle that has mass, electric charge, and colour charge, as well as an additional property called flavour, which describes what type of quark it is (up, down, strange, charm, top, or bottom). Due to an effect known as colour confinement, quarks are never seen on their own. Instead, they form composite particles known as hadrons so that their colour charges cancel out. Hadrons made of one quark and one antiquark are known as mesons, while those made of three quarks are known as baryons. These 'regular' hadrons are well documented and characterized; however, there is nothing in theory to prevent quarks from forming 'exotic' hadrons such as tetraquarks with two quarks and two antiquarks, or pentaquarks with four quarks and one antiquark.

Structure

A wide variety of pentaquarks are possible, with different quark combinations producing different particles. To identify which quarks compose a given pentaquark, physicists use the notation qqqq$−$, where q and $0 b$ respectively refer to any of the six flavours of quarks and antiquarks. The symbols u, d, s, c, b, and t stand for the up, down, strange, charm, bottom, and top quarks respectively, with the symbols of $0 b$, $Λ ψs$, $+ c$, $\overline{q}$, $\overline{q}$, $\overline{u}$ corresponding to the respective antiquarks. For instance a pentaquark made of two up quarks, one down quark, one charm quark, and one charm antiquark would be denoted uudc$\overline{d}$.

The quarks are bound together by the strong force, which acts in such a way as to cancel the colour charges within the particle. In a meson, this means a quark is partnered with an antiquark with an opposite colour charge – blue and antiblue, for example – while in a baryon, the three quarks have between them all three colour charges – red, blue, and green. In a pentaquark, the colours also need to cancel out, and the only feasible combination is to have one quark with one colour (e.g. red), one quark with a second colour (e.g. green), two quarks with the third colour (e.g. blue), and one antiquark to counteract the surplus colour (e.g. antiblue).

The binding mechanism for pentaquarks is not yet clear. They may consist of five quarks tightly bound together, but it is also possible that they are more loosely bound and consist of a three-quark baryon and a two-quark meson interacting relatively weakly with each other via pion exchange (the same force that binds atomic nuclei) in a "meson-baryon molecule".

Mid-2000s
The requirement to include an antiquark means that many classes of pentaquark are hard to identify experimentally – if the flavour of the antiquark matches the flavour of any other quark in the quintuplet, it will cancel out and the particle will resemble its three-quark hadron cousin. For this reason, early pentaquark searches looked for particles where the antiquark did not cancel. In the mid-2000s, several experiments claimed to reveal pentaquark states. In particular, a resonance with a mass of $\overline{s}$ (4.6 σ) was reported by LEPS in 2003, the. This coincided with a pentaquark state with a mass of $\overline{c}$ predicted in 1997. The proposed state was composed of two up quarks, two down quarks, and one strange antiquark (uudd$\overline{b}$). Following this announcement, nine other independent experiments reported seeing narrow peaks from and , with masses between $\overline{t}$ and $\overline{c}$, all above 4 σ. While concerns existed about the validity of these states, the Particle Data Group gave the a 3-star rating (out of 4) in the 2004 Review of Particle Physics. Two other pentaquark states were reported albeit with low statistical significance—the (ddss$1,540 MeV/c2$), with a mass of $1,530 MeV/c2$ and the  (uudd$\overline{s}$), with a mass of $1,522 MeV/c2$. Both were later found to be statistical effects rather than true resonances.

Ten experiments then looked for the, but came out empty-handed. Two in particular (one at BELLE, and the other at CLAS) had nearly the same conditions as other experiments which claimed to have detected the (DIANA and SAPHIR respectively). The 2006 Review of Particle Physics concluded:

"[T]here has not been a high-statistics confirmation of any of the original experiments that claimed to see the ; there have been two high-statistics repeats from Jefferson Lab that have clearly shown the original positive claims in those two cases to be wrong; there have been a number of other high-statistics experiments, none of which have found any evidence for the ; and all attempts to confirm the two other claimed pentaquark states have led to negative results. The conclusion that pentaquarks in general, and the, in particular, do not exist, appears compelling."

The 2008 Review of Particle Physics went even further:

"There are two or three recent experiments that find weak evidence for signals near the nominal masses, but there is simply no point in tabulating them in view of the overwhelming evidence that the claimed pentaquarks do not exist... The whole story—the discoveries themselves, the tidal wave of papers by theorists and phenomenologists that followed, and the eventual 'undiscovery'—is a curious episode in the history of science."

Despite these null results, LEPS results continued to show the existence of a narrow state with a mass of $1,555 MeV/c2$, with a statistical significance of 5.1 σ.

However this 'discovery' was later revealed to be due to flawed methodology (https://www.osti.gov/biblio/21513283-critical-view-claimed-theta-sup-pentaquark).

2015 LHCb results
In July 2015, the LHCb collaboration at CERN identified pentaquarks in the Λ$\overline{u}$→J/ψK$1,860 MeV/c2$p channel, which represents the decay of the bottom lambda baryon (Λ$\overline{c}$) into a J/ψ meson (J/ψ), a kaon (K$3,099 MeV/c2$) and a proton (p). The results showed that sometimes, instead of decaying via intermediate lambda states, the Λ$1,524 MeV/c2$ decayed via intermediate pentaquark states. The two states, named P$0 b$(4380) and P$−$(4450), had individual statistical significances of 9 σ and 12 σ, respectively, and a combined significance of 15 σ – enough to claim a formal discovery. The analysis ruled out the possibility that the effect was caused by conventional particles. The two pentaquark states were both observed decaying strongly to J/ψp, hence must have a valence quark content of two up quarks, a down quark, a charm quark, and an anti-charm quark (up quarkup quarkdown quarkcharm quarkcharm antiquark), making them charmonium-pentaquarks.

The search for pentaquarks was not an objective of the LHCb experiment (which is primarily designed to investigate matter-antimatter asymmetry) and the apparent discovery of pentaquarks was described as an "accident" and "something we've stumbled across" by the Physics Coordinator for the experiment.

Studies of pentaquarks in other experiments

The production of pentaquarks from electroweak decays of Λ$+ c$ baryons has extremely small cross-section and yields very limited information about internal structure of pentaquarks. For this reason, there are several ongoing and proposed initiatives to study pentaquark production in other channels.

It is expected that pentaquarks will be studied in electron-proton collisions in Hall B E12-12-001A and Hall C E2-16-007 experiments at JLab. The major challenge in these studies is a heavy mass of the pentaquark, which will be produced at the tail of photon-proton spectrum in JLab kinematics. For this reason, the currently unknown branching fractions of pentaquark should be sufficiently large to allow pentaquark detection in JLab kinematics. The proposed Electron Ion Collider which has higher energies is much better suited for this problem.

An interesting channel to study pentaquarks in proton-nuclear collisions was suggested by Schmidt & Siddikov (2016). This process has a large cross-section due to lack of electroweak intermediaries and gives access to pentaquark wave function. In the fixed-target experiments pentaquarks will be produced with small rapidities in laboratory frame and will be easily detected. Besides, if there are neutral pentaquarks, as suggested in several models based on flavour symmetry, these might be also produced in this mechanism. This process might be studied at future high-luminosity experiments like After@LHC and NICA.

2019 LHCb results
On 26 March 2019, the LHCb collaboration announced the discovery of a new pentaquark, based on observations that passed the 5-sigma threshold, using a dataset that was many times larger than the 2015 dataset.

Designated Pc(4312)+ (Pc+ identifies a charmonium-pentaquark while the number between parenthesis indicates a mass of about 4312 MeV), the pentaquark decays to a proton and a J/ψ meson. The analyses revealed additionally that the earlier reported observations of the Pc(4450)+ pentaquark actually were the average of two different resonances, designated Pc(4440)+ and Pc(4457)+. Understanding this will require further study.

2022 LHCb results
On 5 July 2022, the LHCb collaboration announced the discovery of a further new pentaquark, with a significance of 15-sigma. Designated PψsΛ(4338)0, its composition is described as udsc$0 b$, representing the first confirmed pentaquark containing a strange quark.

Applications

The discovery of pentaquarks will allow physicists to study the strong force in greater detail and aid understanding of quantum chromodynamics. In addition, current theories suggest that some very large stars produce pentaquarks as they collapse. The study of pentaquarks might help shed light on the physics of neutron stars.

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

title: Pentaquark


text:
1990 Governor General's Awards

The 1990 Governor General's Awards for Literary Merit were prizes awarded to authors in 1990. Each winner of the prize received $10000 and a specially bound edition of their book. The winners were selected by a panel of judges administered by the Canada Council for the Arts.

title: 1990 Governor General's Awards


text:
Concordance

Concordance may refer to:

 * Agreement (linguistics), a form of cross-reference between different parts of a sentence or phrase
 * Bible concordance, an alphabetical listing of terms in the Bible
 * Concordant coastline, in geology, where beds, or layers, of differing rock types form ridges that run parallel to the coast
 * Concordant pair, in statistics
 * Concordance (publishing), a list of words used in a body of work, with their immediate contexts
 * Concordance (genetics), the presence of the same trait in both members of a pair of twins (or set of individuals)
 * Concordance (medicine), involvement of patients in decision-making to improve patient compliance with medical advice
 * Concordance of evidence, in law, science, history, etc.
 * Concordance system, in Swiss politics, the presence of all major parties in the Federal Council
 * Concordance correlation coefficient, in statistics, a measurement of the agreement between two variables
 * Concordance database, a database tailored to legal applications and distributed by LexisNexis
 * Inter-rater reliability, in statistics, the degree to which multiple measurements of the same thing are similar
 * Lambda-CDM model of big-bang cosmology
 * Link concordance, a relation between mathematical links in knot theory

title: Concordance


text:
West Springfield, Massachusetts

West Springfield is a city in Hampden County, Massachusetts, United States. It is part of the Springfield, Massachusetts Metropolitan Statistical Area. The population was 28,835 at the 2020 United States Census. The city is also known as "West Side", in reference to the fact that it is on the western side of the Connecticut River from Springfield, a fact which played a major part in the town's early history.

Early settlement
The area that became known as West Springfield was settled in 1635. The settlers fled to higher ground on the east side of the river and founded Springfield in the aftermath of the great hurricane of 1635. West Springfield was good farm land, so some families did stay on the west side.

Early transportation problems
Other than the trade in beaver skins, economic activity in early colonial Springfield consisted largely of subsistence farming and animal husbandry, with barter being the preferred medium of exchange for neighbors' crops, and locally produced goods. Gristmills and saw mills were also present in the early settlement.

Because the Connecticut River was too wide to be bridged at the time, crossings had to be made by boat. The Hay Place was created between the current town common and East School Street, for people who farmed or mowed on land grants on the west side to leave their crops while they awaited transport back to the eastern side.

By the 1650s some English settlers had begun living full-time on the western side of the river, probably near what is now Riverdale Road, across from the Chicopee River.

Early in that decade, Springfield had made a provision that any able-bodied man (and his work animals) could be required to work up to six eight-hour days on local roads (the barter economy equivalent of an infrastructure tax). In 1666, the west side residents complained about having to work on east side roads while their own were not well taken care of. After considerable dispute, it was determined that the men of the settlement would tend the roads on their own sides of the river.

Parish formation and growing independence
In many ways, the distinction between the church and the state in the early New England town form of government was fuzzy, though religious and secular meetings were held separately and generally led by different people.

For the early settlers of Springfield, attendance at both town meetings and weekly Congregational church services (often both held in the town "meeting house") in the early settlement were mandatory, and this was enforced by fines.

For several decades, West Side residents requested accommodation from the town in the form of a free ferry service to reach the Springfield meeting house on the east side of the river, but were refused by town meeting and even by arbitrators from Northampton and Hadley. In March 1683, Reice Bedortha, his son John, John's wife Lydia, and their newborn Mercy, were drowned on the Connecticut on their way to church when their boat capsized. The west side residents renewed their complaints and began to demand their own church meeting house. On 29 May 1697, the Massachusetts General Court finally approved a separate parish and meeting house for the approximately 200 residents.

West side parishes were also created for Agawam (1696), Feeding Hills (1800), and Holyoke ("North Parish" or "Ireland Parish" named for early Irish settlers John and Mary Riley; 1786).

The Massachusetts Bay Colony passed a law in 1647 requiring the construction of a public school in any town with 50 or more families. In 1706 after two years of petitioning, west side residents were granted funds for the construction of a school (though west side students might have been home-schooled before that time).

In 1707, the west side parish was delegated from Springfield town meeting the right to grant land in its territory.

Independence from Springfield
Given the continuing need to cross the Connecticut River to attend town meetings, and east–west tension over resource allocation, the west side residents petitioned the Massachusetts General Court to be incorporated as a separate town in 1756. After a particularly contentious town meeting in 1773 which bounced between meeting houses on opposite shores and nearly resulted in a year-long government shutdown, proposals for partition were eventually sent from both sides to the colonial legislature. On 23 February 1774, West Springfield was incorporated as a separate town, with territory including what is now Agawam and most of Holyoke.

Another dispute was immediately created when the charter of the town prevented it from taxing the property of Springfield residents within its boundaries. This law was later changed to apply only until such property was sold, but the last such parcel did not become taxable by West Springfield until the 1860s.

American Revolution
West Springfield minutemen participated in the American Revolutionary War beginning on April 20, 1775, the day after the Battles of Lexington and Concord. In 1777, a major contingent of Hessian and British troops were captured at the Battle of Saratoga and transported to Boston (for possible deportation or imprisonment). While encamped in West Springfield, some of the German mercenaries stayed and married into the local population.

Economic conditions after the Revolution led to Shays' Rebellion in Springfield and West Springfield in 1786–1787.

Bridges
Technological advancements allowed the first bridge to be built across the Connecticut River in 1805. It was a toll bridge built on stone pilings; the roadway heaved up and down as it passed over six arch-shaped spans. This bridge was damaged by spring floods in 1814, and after a partial collapse under heavy traffic, was demolished.

In 1816, a replacement bridge opened at Bridge Street. It was destroyed in 1818 by spring ice, despite a valiant attempt to keep it from being washed downstream by tying it to a tree; the cable snapped. A third bridge built on the same foundations, was in use for over 100 years, and known as the "Old Toll Bridge", though tolls were removed in 1873.

The modern Memorial Bridge was opened in 1922; it underwent a major overhaul in the 1990s.

The first North End Bridge opened 1887 with a sturdy metal box-shaped truss (the upper part of the box being suspended above the roadway). In 1923, the tar-sealed wooden decking caught fire, which was made worse by the gas mains the bridge carried. The replacement bridge at this location is still in use.

Old county records indicate a license was given to Benjamin Ashley of West Springfield to operate a ferry across the Connecticut River in 1843, in the Riverdale neighborhood at what are now Ashley Avenue and Ferry Street. A wooden toll bridge was built on this site (at the base of Wayside Avenue—formerly Bridge Street—and Ashley Avenue) in 1847, but burned down in 1903. It was evidently replaced, as the state closed a bridge to Chicopee on this site in 1972, and demolished it in 1987.

Several crossings of the Westfield River were built in the 19th century, but most were destroyed by floods. Several highway bridges were also constructed in the late 20th century.

Natural disasters
The warnings of the Agawam Indians proved true in 1647, 1767, 1801, 1804, and 1818. Civil War-era dikes held back high water in the Agawam (Westfield) River in 1878, but heavy rain flooded the town again in 1927. Both heavy rains and a large snowmelt brought an even more massive flood in 1936, with 8,000 people were displaced in the town of 17,000. The area's bridges survived; the railroad bridge being weighed down by a fully loaded freight train intentionally parked across it. The New England Hurricane of 1938 flooded crops along Riverdale Road and severely damaged the Exposition grounds, causing the fair to close for the season. It also opened a hole in the dike at Mosley Avenue, which was repaired before the rain waters could once again flood the lower section of town. Yet another major flood struck in 1955 from Hurricane Connie and Hurricane Diane arriving five days apart, knocking out the town's drinking water facilities in Southwick and destroying Bear Hole Dam, Piper Reservoir, and Memorial Pool (all of which were rebuilt).

Winter weather has also caused significant damage at times during West Springfield's history. The Great Blizzard of 1888 dropped over 5 ft of snow, with 20 ft drifts. There have also been more recent blizzards in 1978 and 1996.

On June 1, 2011, a tornado touched down in West Springfield, crossed the Connecticut River, and then devastated the City of Springfield, Massachusetts. It damaged densely populated parts of West Springfield, causing two fatalities in the city—including a mother who died while shielding her 15-year-old daughter. U.S. President Barack Obama declared the area surrounding West Springfield and Springfield a federal disaster area.

On October 29, 2011, a snow storm dumped more than ten inches of wet snow on the town and the surrounding area. Snow clung to trees which still had most of their leaves. The result was the falling of trees and limbs on homes, vehicles, powerlines and roadways. It took more than one week for some homes to have power restored.

Agriculture
Agriculture continued to dominate the local economy when market gardening started in the 1830s, concentrating in the Riverdale Road area. These crops were intended to be sent to market for cash, rather than to be used by the farming family for themselves or to barter for other crops. Growing population and improved transportation links increased the size of the potential market; by 1860, West Springfield was using greenhouses and exporting fresh crops to Boston. Agriculture remained an important part of the West Springfield economy for many decades, but land development and economic changes led to a decline, and by the 1940s, it was a minor activity in the town.

The Eastern States Exposition started in 1917 as a reaction against the slow decline of New England agriculture. The annual fall fair is by far West Springfield's largest tourist attraction and one of the largest fairs in the country. The exposition grounds host many events on a year-round basis.

The first Morgan horse was bred in West Springfield in 1789–1790.

Railroads and industrialization

Light manufacturing began to grow in the 19th century, including tanned hides, horse carriages, gunpowder, ceramics, industrial pipes, hats, and boats.

When the Industrial Revolution reached Western Massachusetts in the 19th century, the region's many fast-moving rivers resulted in a mill town boom. Early textile and paper mills were staffed by Irish famine immigrants who nearly doubled their population in the town between 1840 and 1860. Paper manufacturing became a major regional industry, including within the town limits included (mostly clustered on the Westfield River) the Southworth Paper Company (1839), the Agawam Paper Company (1859), the Agawam Canal Company, the Springfield Glazed Paper Company (1882), the Worthy Paper Company (1892), the Mittineague Paper Company (1892, later known as the Strathmore Paper Company and acquired by International Paper )

The Western Railroad opened for freight and passenger service in 1841, connecting West Springfield to Worcester, Boston, the Berkshires, and upstate New York. It would become the Boston and Albany Railroad in 1870. Travel time from Boston to Albany was considerably reduced from the over 40 hours it took by stagecoach in the 1820s. The covered wooden railroad bridge across the Connecticut which opened in 1841, was replaced by the current double-track steel truss railroad bridge in 1874.

West Springfield became a major transportation hub, and the railroad became one of the largest employers in the town for many decades. Repair shops were also built in West Springfield in 1896, and at the peak of operations, there were two major rail yards—one in Mittineague, and one near the present-day Memorial Avenue.

The original horsecar trolley, operated by the Springfield Street Railway, opened in 1877 from Main Street in Springfield to Elm and Park Streets, via Main Street and the old toll bridge at Bridge Street. It was later extended via Westfield Street to (Upper) Church Street. Electrification was completed in 1892–1893, and the river crossing was moved to the original North End Bridge. Over the years, extensions were made to the Holyoke Street Railway (via Riverdale Road, 1895), Tatham (1896) the Woronoco Street Railway (in Westfield, 1899), the Connecticut border via Riverside Park (now Six Flags New England) in Agawam (1900), Feeding Hills (1902), and eventually the Suffield Street Railway in Connecticut (making the Hartford-West Side Line possible, 1905).

The destruction of the old North End Bridge in 1923 saw relocation of the trolley crossing to the modern Memorial Bridge. But trolley passenger service was cut starting in 1924 and by 1936, eliminated. Present-day local and intercity mass transit is provided by Pioneer Valley Transit Authority bus routes, Amtrak, and private bus carriers. Peter Pan Bus Lines is headquartered in Springfield.

Conversion from steam to diesel locomotives shut down the West Springfield repair shop in 1956. With the rise of the automobile, the West Springfield (Mittineague) passenger railroad station closed in 1957. Amtrak service is still available to Springfield, and the central rail yard is still in active use for freight by CSX, the present-day successor of this part of the Boston & Albany.

Rural Free Delivery started delivering postal mail to residents' homes in the late 19th or early 20th century.

A major power plant for the Western Massachusetts Electric Company (now a subsidiary of Northeast Utilities) went online in West Springfield in 1949.

Creation of Holyoke and Agawam
Even more substantial canal and mill development took place in the "North Parish" or "Ireland Parish" of West Springfield, which was favorably located near Hadley Falls. The parish was incorporated as the independent town of Holyoke, Massachusetts, in 1850.

The area mainly south of the Westfield River, including the parishes of Agawam and Feeding Hills, was incorporated as the independent town of Agawam, Massachusetts, in 1855.

Highway construction
U.S. Route 5 (currently, also known as Riverdale Street) was modified to bypass the downtowns of Springfield and West Springfield as new segments were constructed on the West Springfield and Agawam waterfronts in 1938, 1941–1942, and 1952–1953. This resulted in some land takings and cutting off certain neighborhoods from the river, but north–south travel was speeded, and the dike system was reinforced to prevent the flooding of these neighborhoods. The approaches to the North End and Memorial Bridges were modified to accommodate the new traffic patterns.

The Massachusetts Turnpike was constructed from 1955 to 1957. Interstate 91 was constructed over a dozen years, from 1958 to 1970, following considerable controversy over whether it should be placed in West Springfield, as originally planned, or in Springfield, as that city's planners wished.

Interstate 91 planned for West Springfield
The original plan for Interstate 91—detailed in the 1953 Master Highway Plan for the Springfield, Massachusetts, Metropolitan Area—called for Interstate 91 to occupy an enlarged U.S. Route 5 in West Springfield—the route which had, historically, been used to reach West Springfield and Springfield from both the north and the south. Between 1953 and 1958, Riverdale Road was widened in places, added on to, and numerous businesses were closed and moved back, or to other parts of West Springfield to make way for Interstate 91, which was planned to connect with Springfield via numerous bridges. The original plan for I-91 would have likely benefitted West Springfield, which already had U.S. 5 passing through, causing travelers to patronize many of West Springfield's businesses.

In 1958, however, Springfield's city planners campaigned vociferously for Interstate 91 to occupy Springfield's riverfront. Their reasoning at the time was that Springfield, being a more populous city than West Springfield, should have a major highway routed through it. Indeed, Springfield's 1958 city planners advocated that the construction of I-91 on Springfield's riverfront would catalyze economic growth comparable to that experienced during the great railroad expansion of the mid-19th century.

Massachusetts highway officials decided to route the highway on the Springfield side, finding that a short existing section of US 5 through West Springfield that was built in 1952–1953 failed to meet Interstate design standards. On the east bank of the river, Springfield got an elevated highway, as opposed to the planned ground-level highway in West Springfield.

After Interstate 91 was constructed in Springfield, that city did not experience anything like the prosperity boom predicted by its city planners in 1958. I-91's construction in Springfield coincided with the beginning of that city's four decades of decline. Unlike West Springfield's U.S. 5, Springfield's I-91 was constructed in an area where there had never been highway traffic or businesses that catered to such traffic. Due to I-91's proximity to both Springfield's densely built downtown and the city's riverfront, there has never been enough space in Springfield to build more than a few of these businesses. Thus Springfield never received the economic benefit that West Springfield would have, according to a 2009 assessment by the UMass School of Urban Design.

Geography
According to the United States Census Bureau, the city has a total area of 17.5 square miles (45.4 km$2$), of which 16.8 square miles (43.4 km$2$) is land and 0.8 square mile (2.0 km$2$) (4.50%) is water. It is on the west side of the Connecticut River, across from Springfield, and on the north side of the Westfield River, above Agawam.

West Springfield is bordered on the west by linear cliffs of volcanic trap rock known as East Mountain. They are part of the Metacomet Ridge, a mountainous trap rock ridgeline that stretches from Long Island Sound to nearly the Vermont border. Both are traversed by the 110 mi Metacomet-Monadnock Trail.

West Springfield is located 27 mi away from Hartford, 90 mi away from Boston, 85 mi from Albany, 48 mi away from Pittsfield, 139 mi from New York City and 6 mi miles from Six Flags.

Other major geographical features include:
 * Bagg Brook
 * Block Brook (named after a distinctive bridge on what is now Westfield Street)
 * Bear Hole – an approximately 1700 acre wooded area that includes Bear Hole Reservoir. The reservoir, which is very shallow, supplies a minimal amount of the town's drinking water. The area supports a diversified biological environment; including Great Horned Owl, White Tailed Deer, hawk as well as vernal pools. Hiking, dog walking and nature watching are popular activities year round. The Pioneer Valley Railroad has a railway established along the westernmost perimeter. This is a very low speed railway that hauls primarily commercial freight to local businesses. In 2021, the Massachusetts Audubon Society and the Massachusetts Department of Conservation and Recreation purchased a permanent conservation easement for the property (which extends into Holyoke) from the Town of West Springfield, which owns the land for drinking water protection purposes.  The origin of the name is unclear; the first known written reference is a deed showing ownership by Rufus S. Payne in 1838. From 1890 to 1906 it was home to the Bear Hole Resort, including a restaurant and a bear in a cage.
 * Crowfoot Brook
 * Goldine Brook
 * Mittineague Park
 * Pawcatuck Brook – from morphemes in local Indian language: pauqua – clear, transparent, or pegwa – shallow; tuck – river
 * Piper Brook
 * Town Common – Formerly used for the town meeting house, church, and militia, the current tree-lined park was laid out in 1866 by Edward Parsons and other townspeople. The surrounding streets were at this time renamed from "Broadway" to "Park Street" and "Park Avenue".
 * White Church Hill – The church which still remains there replaced the Old Meeting House (which was on the Common) as the town's main church and meeting hall in 1802.  It was supplemented by a second congregational church, again on the common, in 1872.
 * Schoolhouse Brook
 * Tannery Brook

Neighborhoods

 * Amostown, West Springfield - Named for Amos Taylor, a settler in the 18th century
 * Ashleyville, West Springfield - Named after the Ashley family and located near Interstate 91 and Riverdale Street
 * Cayenne, West Springfield - Named by resident Russell H. Pepper, as a parody of his own name and located near Morgan Road and Riverdale Street
 * Downtown West Springfield - Area surrounding Elm St, Park St, and Van Deene Ave. Also known as Center.
 * Memorial, West Springfield - Area closest to Memorial Ave and the Big E
 * Merrick, West Springfield - Named after the Merrick family, prominent land owners in the most densely settled part of town.
 * Mittineague - An Indian name for the same place, and located on the north bank of the Westfield River
 * Paucatuck, West Springfield (archaic) - area near Paucatuck Brook, west of Tatham
 * Tatham, West Springfield (Tawtum, Tattom) - An Indian name for the same place, located northwest and west of Mittineague
 * Ramapogue, West Springfield (archaic) - the area west of the Common and east of the hills

Roads
Numbered routes passing through the town include:
 * Interstate 91 (perpendicular to Route 5 after crossing from Springfield, through the northeast quadrant of town)
 * Massachusetts Turnpike (I-90)
 * U.S. Route 5 (Riverdale Street)
 * U.S. Route 20 (Westfield Street and downtown)
 * Massachusetts Route 147 (Memorial Avenue)

Airport
There are no airports in West Springfield, the nearest airport with commercial passenger flights is Bradley International Airport in Windsor Locks, Connecticut. The nearest general aviation airports are Westover Metropolitan Airport in Chicopee, and Westfield-Barnes Regional Airport, and there are helipads in Agawam and Springfield.

Transit
As for transportation, several bus lines on the Pioneer Valley Transit Authority bus system service West Springfield from routes crossing over from Springfield:

 * Red 14 – Feeding Hills/Springfield
 * Red 10 – West Springfield/Westfield State University
 * Purple 20 – Holyoke/Springfield via Holyoke Mall – Riverdale
 * Red 24 – Essex/Appleton via Cabot/Sargent/Holyoke Hospital (Saturday only)

Demographics
As of the census of 2010, there were 28,391 people, 11,757 households, and 7,117 families residing in the city. The population density was 1,665.7 PD/sqmi. There were 12,259 housing units at an average density of 731.9 /sqmi. The racial makeup of the city was 90.68% White, 3.30% African American, 0.22% Native American, 1.97% Asian, 0.04% Pacific Islander, 2.94% from other races, and 2.10% from two or more races. 5.75% of the population were Hispanic or Latino of any race.

There were 11,823 households, out of which 27.0% had children under the age of 18 living with them, 44.7% were married couples living together, 11.4% had a female householder with no husband present, and 39.8% were non-families. 34.0% of all households were made up of individuals, and 11.2% had someone living alone who was 65 years of age or older. The average household size was 2.33 and the average family size was 3.02.

In the city, the population was spread out, with 23.4% under the age of 18, 7.8% from 18 to 24, 29.6% from 25 to 44, 23.2% from 45 to 64, and 15.9% who were 65 years of age or older. The median age was 38 years. For every 100 females, there were 95.2 males. For every 100 females age 18 and over, there were 91.2 males.

The median income for a household in the city was $40,266, and the median income for a family was $50,282. Males had a median income of $38,082 versus $28,079 for females. The per capita income for the city was $20,982. About 8.7% of families and 11.9% of the population were below the poverty line, including 20.0% of those under age 18 and 7.3% of those age 65 or over.

 * Population table source: Populations for 1800, 1810, and 1830 were not available online from this source.

Government
West Springfield was originally governed by an open town meeting, and then a representative town meeting starting in 1922. The town meetings were presided over by the town moderator. The city is now governed by a mayor and town council starting on April 1, 2000. West Springfield is one of thirteen Massachusetts municipalities that have applied for, and been granted, city forms of government but wish to retain "The town of" in their official names. The first mayor who held office was Edward Gibson, an independent, who stayed in office for five consecutive terms. Republican Greg Neffinger, was elected mayor on November 8, 2011, but lost his bid for re-election to Republican Edward C Sullivan in 2013. The current mayor, Independent Will Reichelt was elected to his first term in 2015. He was easily re-elected in 2017, and won a third term unopposed in 2021.

 * 2012

Town services
West Springfield has its own school district, police department, fire department, library, department of public works, health department, and department of parks and recreation.

West Springfield Fire Department was created in 1883 and current has 11 apparatus serving out of one fire hall.

The school district has controlled by an independent school committee since 1827 and maintains a central high school, middle school, and a number of elementary schools.

The first town funds budgeted for police enforcement were in 1877; the force has gradually become professionalized, motorized, and equipped with and dedicated offices and jail facilities.

The Department of Public Works provides water utility service to the town, drawing from four wells in the Great Brook section of Barne's Aquifer, in Southwick. The Bear Hole Reservoir and Bear Hole Water Treatment Plant are no longer used. The well-water treatment plant (at 506 Shaker Road, Westfield, on the Southwick border) uses 240,000 pounds of granular activated carbon to filter PFAS and the pesticides ethylene dibromide and dichloropropane from the well water. The pesticides were used on nearby fields (currently used for tobacco and hay). The city also has two interconnections (on Piper Road and Rogers Ave) with the Springfield Water and Sewer Commission for a backup supply, fed from Cobble Mountain Reservoir via a treatment facility in Westfield. West Springfield water tanks are located at 537 Rogers Ave, at the end of Goldenview Drive, and on Prospect Hill near 733 Bernie Ave.

Library
Semi-public "subscription" libraries operated from 1775 to 1807 and 1810–1840. The current collection began in 1854, but was later made free to the public. The West Springfield Public Library was established in 1864. It moved from the town hall to a dedicated building in 1915, constructed with a donation from Andrew Carnegie. An addition was opened in 1959. In fiscal year 2008, the city of West Springfield spent 0.91% ($765,760) of its budget on its public library—approximately $27 per person, per year ($35.58 adjusted for inflation to 2022).

Education
The West Springfield Public Schools district operates nine schools. The town also has a charter school, a Catholic school, and two schools for special education. The town's public school athletic teams are called the "Terriers".

Public district schools

 * Cowing School (preschool and other students who are not able to attend regular school)
 * John Ashley School (PreK-K)
 * Phillip G. Coburn Elementary (K-5)
 * John R. Fausey Elementary (1-5)
 * Memorial Elementary (1-5)
 * Mittineague Elementary (1-5)
 * Tatham Elementary (1-5)
 * West Springfield Middle School (6-8)
 * West Springfield High School (9-12)

Other schools

 * Lower Pioneer Valley Educational Collaborative (special education, jointly operated with neighboring school districts)
 * St. Thomas the Apostle (K-8) - Catholic school
 * Hampden Charter School of Science West (6-11)
 * May Center School for Autism and Developmental Disabilities (private)

School buildings and closings
Coburn Elementary School was built in 1923–1924 as West Springfield Junior High School and added to in 1928–1929, operating as a grade 7–8 junior high school. Cowing School was opened as the town's first stand-alone high school in 1915, and when the new high school on Piper Rd. opened in 1956 it became Cowing Junior High School, serving grades 7–8 as well. As West Springfield moved to the middle-school format, Coburn and Cowing assumed their current functions. Three elementary schools—John Ashley, Memorial, and Tatham—were opened in 1952 to accommodate the post-war baby boom. Additionally, John R. Fausey School was built in 1960 and added to in 1962. West Springfield High School was moved from the Cowing building in 1956 to Piper Road. A significant addition opened in 1966. A new West Springfield High School was built to the north of the old building and was opened in 2014. The old building was demolished. In 2019 modular buildings were added to Tatham Elementary because of increased enrollments and little space. The modulars currently hold 5th and 3rd grade classes. After the construction of the first high school on Piper Road, Cowing School was originally slated for demolition, but the new high school had been under-built for the surging student population, and 9th graders were moved to Cowing from 1960 until the high school addition opened in 1966. From 1966 to 1981, Cowing School served as a junior high school.

With the passage of the Proposition 2-1/2 referendum in 1980, the school budget was tightened, priorities were reexamined, and numerous schools were closed or consolidated in the fall of 1981.
 * John Ashley School closed but reopened in 1987, due to increased district enrollment, as the town-wide center for kindergarten and early childhood classes.
 * Main Street School closed and was repurposed as apartments.
 * Park Avenue School closed and was repurposed as an office building.
 * Old Tatham School closed and was repurposed as apartments.
 * Margaret C. McDonough (formerly Kings Highway) School closed and was repurposed as apartments.
 * William A. Cowing Junior High School (formerly the High School from 1915 to 1956) was designated an elementary school.

Before the opening of West Springfield Middle School in 1998, the town struggled with increased enrollment.
 * In 1987, John Ashley School welcomed the future class of 2000, consolidating all kindergartens in one building along with the early childhood program.
 * In 1990, Piper Road School opened in a wing of the high school, housing all the town's 6th graders, and creating the grade 1–5 elementary school structure that remains in most buildings. With the opening of the middle school, this space reverted to the high school.

West Springfield Middle School opened in 1998 for grades 6–8.
 * West Springfield Junior High School was designated an elementary school and named the Philip G. Coburn School in memory of a well-loved townsman who was a former educator and longtime editor of the West Springfield Record.
 * William A. Cowing Elementary School was closed and sections of the building were used for different educational purposes until the district adopted full-day kindergarten and early childhood (Pre-K) students were assigned to Cowing School.
 * In 2014, a new West Springfield High School was built and the 1956 building was demolished.
 * In 2019, modular buildings were added to Tatham Elementary to increase space. The buildings are used as fifth and third grade classrooms.

Economy
West Springfield is considered to be part of the Springfield-Hartford Knowledge Corridor.

With easy access to the north–south Interstate 91, east–west Massachusetts Turnpike, and various freight railroads, West Springfield is sometimes called the "crossroads of New England".

The Riverdale Road corridor is a major regional shopping center, with a number of "big box" retail stores and car dealerships.

The Eastern States Exposition is a major seasonal employer.

Tourist attractions

 * The Josiah Day House, which is believed to be the oldest known brick saltbox style house in the United States.
 * The Eastern States Exposition (the "Big E"), a large annual fair
 * The 110 mi Metacomet-Monadnock Trail (a hiking trail) passes through the western part of West Springfield on the East Mountain and Provin Mountain ridgeline.

There are only a few parks in the city:
 * Mittineague Park
 * Veterans Field – football field and two baseball diamonds
 * Cook's Park
 * Town Common between Park Street & Park Ave from Elm to Main Streets
 * Ohio Field
 * Old Tatham Field
 * New Tatham Field
 * Clark Field

Media
There is only one local newspaper, the weekly (every Thursday) town newspaper West Springfield Record was founded in 1953 with circulation about 5,500. All other print media are regional papers such as the Republican of Springfield. Television and radio service originates from other cities outside of West Springfield.

West Springfield is one of the two cities (along with Portland, Oregon), that inspired The Simpsons’ hometown of Springfield, though the name is based on Springfield, Oregon.

ABC's short-lived 2004–2005 sitcom Complete Savages used West Springfield as its setting.

Sports
West Springfield was the longtime home of the Springfield Indians from 1926 to 1972 with some spurts in between including the 1940s when the Eastern States Coliseum was used for war surplus during World War II. After the Springfield Civic Center was built downtown, the Indians relocated there for the 1972-73 AHL Season. The ice plant at the Coliseum was in use until 1990.

The Massachusetts Twisters of the American Indoor Soccer League played at the Coliseum as well.

Notable people

 * Angelo Bertelli, football player, first Heisman Trophy winner for Notre Dame
 * Amo Bessone, coach, Michigan State hockey (1966 National Champion) and United States Hockey Hall of Fame inductee, 1992
 * Peter Bessone, United States Hockey Hall of Fame inductee, 1978
 * William Bourque, Canadian auto racer living in West Springfield; died during first AAA Championship race at Indianapolis Motor Speedway
 * Chris Capuano, Major League Baseball pitcher
 * Tim Daggett, gymnast, Olympic gold medalist
 * Harry Dalton, MLB general manager
 * Luke Day, Revolutionary War captain, leader of Shays' Rebellion
 * Leo Durocher, Hall of Fame Major League Baseball player and manager
 * Eugene Grazia, member of 1960 U.S. hockey gold medal team
 * Horace A. Moses, founder of Strathmore (Mittineague) Paper Company (1892) and West Springfield Trust Company (1919), local philanthropist
 * Joe Ragland, Turkish League professional basketball player
 * Vic Raschi, Major League Baseball pitcher, six-time World Series champion
 * Brian Scully, television writer and producer for The Simpsons and Family Guy
 * Mike Scully, former co-producer of The Simpsons
 * Stass Shpanin, contemporary visual artist included in Guinness Book of World Records as Youngest Professional

title: West Springfield, Massachusetts