Pappos Ben Yehuda
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Pappus of Alexandria (; ; AD) was a Greek mathematician of
late antiquity Late antiquity marks the period that comes after the end of classical antiquity and stretches into the onset of the Early Middle Ages. Late antiquity as a period was popularized by Peter Brown (historian), Peter Brown in 1971, and this periodiza ...
known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for
Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
. Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in
Alexandria Alexandria ( ; ) is the List of cities and towns in Egypt#Largest cities, second largest city in Egypt and the List of coastal settlements of the Mediterranean Sea, largest city on the Mediterranean coast. It lies at the western edge of the Nile ...
, where he worked as a
mathematics teacher In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although r ...
to higher level students, one of whom was named Hermodorus.Pierre Dedron, J. Itard (1959) ''Mathematics And Mathematicians'', Vol. 1, p. 149 (trans.
Judith V. Field Judith Veronica Field (born 1943) is a British historian of science with interests in mathematics and the impact of science in art, an honorary visiting research fellow in the Department of History of Art of Birkbeck, University of London, forme ...
) (Transworld Student Library, 1974)
The ''Collection'', his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, and
mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
. Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to some, he stands out as a remarkable exception and, to others, as an exemplar of ills that halted the progress of Greek science. In many respects, his fate strikingly resembles that of Diophantus', originally of limited importance but becoming very influential in the late
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
and
Early Modern The early modern period is a Periodization, historical period that is defined either as part of or as immediately preceding the modern period, with divisions based primarily on the history of Europe and the broader concept of modernity. There i ...
periods.


Dating

In his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time (but see below) when he himself wrote. If no other date information were available, all that could be known would be that he was later than
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
(died c. 168 AD), whom he quotes, and earlier than
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
(born ), who quotes him. The 10th century ''
Suda The ''Suda'' or ''Souda'' (; ; ) is a large 10th-century Byzantine Empire, Byzantine encyclopedia of the History of the Mediterranean region, ancient Mediterranean world, formerly attributed to an author called Soudas () or Souidas (). It is an ...
'' states that Pappus was of the same age as
Theon of Alexandria Theon of Alexandria (; ; ) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on works by Euclid and Ptolemy. His daughter Hypatia also won fame as a mathema ...
, who was active in the reign of Emperor
Theodosius I Theodosius I ( ; 11 January 347 – 17 January 395), also known as Theodosius the Great, was Roman emperor from 379 to 395. He won two civil wars and was instrumental in establishing the Nicene Creed as the orthodox doctrine for Nicene C ...
(372–395). A different date is given by a marginal note to a late 10th-century manuscript (a copy of a chronological table by the same Theon), which states, next to an entry on Emperor
Diocletian Diocletian ( ; ; ; 242/245 – 311/312), nicknamed Jovius, was Roman emperor from 284 until his abdication in 305. He was born Diocles to a family of low status in the Roman province of Dalmatia (Roman province), Dalmatia. As with other Illyri ...
(reigned 284–305), that "at that time wrote Pappus". However, a verifiable date comes from the dating of a solar eclipse mentioned by Pappus himself. In his commentary on the ''
Almagest The ''Almagest'' ( ) is a 2nd-century Greek mathematics, mathematical and Greek astronomy, astronomical treatise on the apparent motions of the stars and planetary paths, written by Ptolemy, Claudius Ptolemy ( ) in Koine Greek. One of the most i ...
'' he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi in 1068 after
Nabonassar Nabû-nāṣir was the king of Babylon from 747 to 734 BC. He deposed a foreign Chaldean usurper named Nabu-shuma-ishkun, bringing native rule back to Babylon after twenty-three years of Chaldean rule. His reign saw the beginning of a new era ...
". This works out as 18 October 320, and so Pappus must have been active around 320.


Works

The great work of Pappus, in eight books and titled ''Synagoge'' or ''Collection'', has not survived in complete form: the first book is lost, and the rest have suffered considerably. The ''
Suda The ''Suda'' or ''Souda'' (; ; ) is a large 10th-century Byzantine Empire, Byzantine encyclopedia of the History of the Mediterranean region, ancient Mediterranean world, formerly attributed to an author called Soudas () or Souidas (). It is an ...
'' enumerates other works of Pappus: ''Χωρογραφία οἰκουμενική'' ('' Chorographia oikoumenike'' or ''Description of the Inhabited World''), a commentary on the thirteen books of
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
's ''
Almagest The ''Almagest'' ( ) is a 2nd-century Greek mathematics, mathematical and Greek astronomy, astronomical treatise on the apparent motions of the stars and planetary paths, written by Ptolemy, Claudius Ptolemy ( ) in Koine Greek. One of the most i ...
'' (of which the part on books 5 and 6 survives), ''Ποταμοὺς τοὺς ἐν Λιβύῃ'' (''The Rivers in Libya''), and ''Ὀνειροκριτικά'' (''The Interpretation of Dreams''). Pappus himself mentions another commentary of his own on the ''Ἀνάλημμα'' (''
Analemma In astronomy, an analemma (; ) is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same Solar time#Mean solar time, mean solar time over the course of a year. The change of position is a result ...
'') of
Diodorus of Alexandria Diodorus of Alexandria or Diodorus Alexandrinus was a gnomonicist, astronomer and a pupil of Posidonius. Writings He wrote the first discourse on the principles of the sundial, known as ''Analemma''. a commentary on this having later been wri ...
. Pappus also wrote commentaries on
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's ''Elements'' (of which fragments are preserved in
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
and the
Scholia Scholia (: scholium or scholion, from , "comment", "interpretation") are grammatical, critical, or explanatory comments – original or copied from prior commentaries – which are inserted in the margin of the manuscript of ancient a ...
, while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy's ''Ἁρμονικά'' (''Harmonika'').
Federico Commandino Federico Commandino (1509 – 5 September 1575) was an Italian humanism, humanist and mathematician. Born in Urbino, he studied at Padua and then at Ferrara, where he received his doctorate in medicine under Antonio Musa Brassavola. He had numer ...
translated the ''Collection'' of Pappus into Latin in 1588. The German classicist and mathematical historian Friedrich Hultsch (1833–1908) published a definitive three-volume presentation of Commandino's translation with both the Greek and Latin versions (Berlin, 1875–1878). Using Hultsch's work, the Belgian mathematical historian Paul ver Eecke was the first to publish a translation of the ''Collection'' into a modern European language; his two-volume, French translation has the title ''Pappus d'Alexandrie. La Collection Mathématique.'' (Paris and Bruges, 1933).


''Collection''

Pappus's ''Collection'' contains an account, systematically arranged, of the most important results obtained by his predecessors and notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively.
Heath A heath () is a shrubland habitat found mainly on free-draining infertile, acidic soils and is characterised by open, low-growing woody vegetation. Moorland is generally related to high-ground heaths with—especially in Great Britain—a coole ...
considered the systematic introductions to the various books valuable, for they set forth clearly an outline of the contents and the general scope of the subjects to be treated. In these introductions, the style of Pappus's writing is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions. Heath also found his characteristic exactness made his ''Collection'' "a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us". The surviving portions of ''Collection'' can be summarized as follows.


Book I

Book I has been lost. Book I, like Book II, may have been concerned with arithmetic, as Book III is clearly introduced as beginning a new subject.


Book II

The whole of Book II (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) discusses a method of multiplication from an unnamed book by
Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
. The final propositions deal with multiplying together the numerical values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to and .


Book III

Book III contains geometrical problems, plane and solid. It may be divided into five sections: # On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by
Hippocrates of Chios Hippocrates of Chios (; c. 470 – c. 421 BC) was an ancient Greek mathematician, geometer, and astronomer. He was born on the isle of Chios, where he was originally a merchant. After some misadventures (he was robbed by either pirates or ...
to the former. Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one. # On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure. This serves as an introduction to a general theory of means, of which Pappus distinguishes ten kinds, and gives a table representing examples of each in whole numbers. # On a curious problem suggested by Euclid I. 21. # On the inscribing of each of the five regular polyhedra in a sphere. Here Pappus observed that a
regular dodecahedron A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...
and a
regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...
could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle. This observation has been generalised to higher dimensional dual polytopes. # An addition by a later writer on another solution of the first problem of the book.


Book IV

Of Book IV the title and preface have been lost, so that the program has to be gathered from the book itself. At the beginning is the well-known generalization of Euclid I.47 (
Pappus's area theorem Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after t ...
), then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. This and several other propositions on contact, e.g. cases of circles touching one another and inscribed in the figure made of three semicircles and known as
arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...
("shoemakers knife") form the first division of the book; Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes (already mentioned in Book I as supplying a method of doubling the cube), and the curve discovered most probably by
Hippias of Elis Hippias of Elis (; ; late 5th century BC) was a Greek sophist, and a contemporary of Socrates. With an assurance characteristic of the later sophists, he claimed to be regarded as an authority on all subjects, and lectured on poetry, grammar, his ...
about 420 BC, and known by the name, τετραγωνισμός, or
quadratrix In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circ ...
. Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time. The area of the surface included between this curve and its base is found – the first known instance of a quadrature of a curved surface. The rest of the book treats of the
trisection of an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
, and the solution of more general problems of the same kind by means of the quadratrix and spiral. In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the focus and directrix.


Book V

In Book V, after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus's treatise on this subject), and of the volumes of different solid figures which have all the same superficial area, and, lastly, a comparison of the five regular solids of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.


Book VI

According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "
Little Astronomy ''Little Astronomy'' ( ) is a collection of minor works in Ancient Greek mathematics and astronomy dating from the 4th to 2nd century BCE that were probably used as an astronomical curriculum starting around the 2nd century CE. In the astronomy o ...
" (Μικρὸς Ἀστρονομούμενος), i.e. works other than the ''
Almagest The ''Almagest'' ( ) is a 2nd-century Greek mathematics, mathematical and Greek astronomy, astronomical treatise on the apparent motions of the stars and planetary paths, written by Ptolemy, Claudius Ptolemy ( ) in Koine Greek. One of the most i ...
''. It accordingly comments on the ''Sphaerica'' of
Theodosius Theodosius ( Latinized from the Greek "Θεοδόσιος", Theodosios, "given by god") is a given name. It may take the form Teodósio, Teodosie, Teodosije etc. Theodosia is a feminine version of the name. Emperors of ancient Rome and Byzantium ...
, the ''Moving Sphere'' of
Autolycus In Greek mythology, Autolycus (; ) was a robber who had the power to metamorphose or make invisible the things he stole. He had his residence on Mount Parnassus and was renowned among men for his cunning and oaths. Family There are a number of d ...
, Theodosius's book on ''Day and Night'', the treatise of Aristarchus '' On the Size and Distances of the Sun and Moon'', and Euclid's ''Optics and Phaenomena''.


Book VII

Since
Michel Chasles Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...
cited this book of Pappus in his history of geometric methods, it has become the object of considerable attention. The preface of Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem. Pappus then enumerates works of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
,
Apollonius Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...
,
Aristaeus Aristaeus (; ''Aristaios'') was the mythological culture hero credited with the discovery of many rural useful arts and handicrafts, including bee-keeping; He was the son of the huntress Cyrene and Apollo. ''Aristaeus'' ("the best") was a cu ...
and
Eratosthenes Eratosthenes of Cyrene (; ;  – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...
, thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation. With the mention of the ''Porisms'' of Euclid we have an account of the relation of
porism A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an in ...
to theorem and problem. In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs one of one set and one of another of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position); (b) the theorems which were rediscovered by and named after
Paul Guldin Paul Guldin (born Habakkuk Guldin; 12 June 1577 (Mels) – 3 November 1643 (Graz)) was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. (This theo ...
, but appear to have been discovered by Pappus himself. Book VII also contains # under the head of the ''De Sectione Determinata'' of Apollonius, lemmas which, closely examined, are seen to be cases of the involution of six points; # important lemmas on the ''Porisms'' of Euclid, including what is called
Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
; # a lemma upon the ''Surface Loci'' of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a
conic A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...
, and is followed by proofs that the conic is a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
,
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, or
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
according as the constant ratio is equal to, less than or greater than 1 (the first recorded proofs of the properties, which do not appear in Apollonius). Chasles's citation of Pappus was repeated by
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...
and Dirk Struik. In Cambridge, England, John J. Milne gave readers the benefit of his reading of Pappus. In 1985 Alexander Jones wrote his thesis at
Brown University Brown University is a Private university, private Ivy League research university in Providence, Rhode Island, United States. It is the List of colonial colleges, seventh-oldest institution of higher education in the US, founded in 1764 as the ' ...
on the subject. A revised form of his translation and commentary was published by Springer-Verlag the following year. Jones succeeds in showing how Pappus manipulated the
complete quadrangle In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six ...
, used the relation of
projective harmonic conjugate In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: :Given three collinear points , let be a point not lying on their join and le ...
s, and displayed an awareness of
cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...
s of points and lines. Furthermore, the concept of
pole and polar In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into i ...
is revealed as a lemma in Book VII.


Book VIII

Book VIII principally treats mechanics, the properties of the center of gravity, and some mechanical powers. Interspersed are some propositions on pure geometry. Proposition 14 shows how to draw an ellipse through five given points, and Prop. 15 gives a simple construction for the axes of an ellipse when a pair of
conjugate diameters In geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate ...
are given.


Legacy

Pappus's ''Collection'' was relatively unknown in medieval Europe, but exerted great influence on 17th-century mathematics after being translated to Latin by
Federico Commandino Federico Commandino (1509 – 5 September 1575) was an Italian humanism, humanist and mathematician. Born in Urbino, he studied at Padua and then at Ferrara, where he received his doctorate in medicine under Antonio Musa Brassavola. He had numer ...
.
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
's ''
Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
'' and Pappus's ''Collection'' were the two major sources of Viète's ''Isagoge in artem analyticam'' (1591). The Pappus's problem and its generalization led Descartes to the development of
analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...
.
Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
also developed his version of analytic geometry and his method of
Maxima and Minima In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
from Pappus's summaries of
Apollonius Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...
's lost works ''Plane Loci'' and ''On Determinate Section''. Other mathematicians influenced by Pappus were
Pacioli Luca Bartolomeo de Pacioli, Order of Friars Minor, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Order of Friars Minor, Franciscan friar, collaborator with Leonardo da Vinci, and an early c ...
,
da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
,
Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of p ...
, van Roomen, Pascal,
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: People * Newton (surname), including a list of people with the surname * ...
, Bernoulli,
Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
,
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
,
Gergonne Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becaus ...
, Steiner and
Poncelet The poncelet (symbol p) is an obsolete unit of power, once used in France and replaced by (ch, metric horsepower). The unit was named after Jean-Victor Poncelet.François Cardarelli, ''Encyclopaedia of Scientific Units, Weights and Measures: T ...
.


See also

*
Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
*
Pappus's centroid theorem In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The ...
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Pappus chain In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction The arbelos is defined by two circles, and , which are tangent at the point and where is ...
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Pappus configuration In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration is named after Pappus of A ...
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Pappus graph In the mathematical field of graph theory, the Pappus graph is a bipartite, 3- regular, undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient ...


Notes


References

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Further reading

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External links

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Scans of Hultsch's Edition of Pappus of Alexandria's Collection
at wilbourhall.org

(Bibliotheca Augustana) *
"Pappus"
''Columbia Electronic Encyclopedia'', Sixth Edition at Answer.com.

at MathPages
Pappus's work on the Isoperimetric Problem
a
Convergence
{{DEFAULTSORT:Pappus of Alexandria Roman-era Alexandrians Ancient Greek geometers 4th-century Greek writers 290s births 350s deaths Year of birth unknown Year of death unknown 4th-century mathematicians