Pappus Of Alexandria
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Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity 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, pro ...
. Nothing is known of his life, other than what can be found in his own writings: that he had a son named Hermodorus, and was a teacher in
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
.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, form ...
) (Transworld Student Library, 1974)
''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, including
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
,
doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
,
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s and
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
.


Context

Pappus was active in the 4th century AD. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. "How far he was above his contemporaries, how little appreciated or understood by them, is shown by the absence of references to him in other Greek writers, and by the fact that his work had no effect in arresting the decay of mathematical science,"
Thomas Little Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translat ...
writes. "In this respect the fate of Pappus strikingly resembles that of
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
."


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 (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
(died c. 168 AD), whom he quotes, and earlier than
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
(born ), who quotes him. The 10th century ''
Suda The ''Suda'' or ''Souda'' (; grc-x-medieval, Σοῦδα, Soûda; la, Suidae Lexicon) is a large 10th-century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Soudas (Σούδας) or Souidas ...
'' states that Pappus was of the same age as
Theon of Alexandria Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς;  335 – c. 405) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on wor ...
, who was active in the reign of Emperor
Theodosius I Theodosius I ( grc-gre, Θεοδόσιος ; 11 January 347 – 17 January 395), also called Theodosius the Great, was Roman emperor from 379 to 395. During his reign, he succeeded in a crucial war against the Goths, as well as in two ...
(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 (; la, Gaius Aurelius Valerius Diocletianus, grc, Διοκλητιανός, Diokletianós; c. 242/245 – 311/312), nicknamed ''Iovius'', was Roman emperor from 284 until his abdication in 305. He was born Gaius Valerius Diocles ...
(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-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
'' 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'' (; grc-x-medieval, Σοῦδα, Soûda; la, Suidae Lexicon) is a large 10th-century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Soudas (Σούδας) or Souidas ...
'' enumerates other works of Pappus: ''Χωρογραφία οἰκουμενική'' ('' Chorographia oikoumenike'' or ''Description of the Inhabited World''), commentary on the four books of
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
'', ''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 mean solar time, as that position varies over the course of a year. The diagram will resemble a figure ei ...
'') 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 w ...
. Pappus also wrote commentaries on
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''Elements'' (of which fragments are preserved in
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
and the
Scholia Scholia (singular scholium or scholion, from grc, σχόλιον, "comment, interpretation") are grammatical, critical, or explanatory comments – original or copied from prior commentaries – which are inserted in the margin of th ...
, 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 humanist and mathematician. Born in Urbino, he studied at Padua and at Ferrara, where he received his doctorate in medicine. He was most famous for his central role as translator o ...
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 Paul-Louis ver Eecke (23 February 1867 – 14 October 1959) was a Belgian mining engineer and historian of Greek mathematics. He produced influential French translations of the mathematical works of ancient Greece, including those of Archimedes, ...
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''

The characteristics of Pappus's ''Collection'' are that it contains an account, systematically arranged, of the most important results obtained by his predecessors, and, secondly, notes explanatory of, or extending, previous discoveries. These discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the various books as valuable, for they set forth clearly an outline of the contents and the general scope of the subjects to be treated. From these introductions one can judge of the style of Pappus's writing, which 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. We can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. 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 ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
. 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 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 ( grc-gre, Ἱπποκράτης ὁ Χῖος; c. 470 – c. 410 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 misadve ...
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 dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
and a
regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...
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. 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), 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 Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ...
, the
conchoid of Nicomedes In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points ...
(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 (; el, Ἱππίας ὁ Ἠλεῖος; 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, ...
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 Ehrenfried Walther von Tschirnhaus, E. W. Tschirnhaus, ...
. 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 a ...
, 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. 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 ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere. According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "Lesser Astronomical Works" (Μικρὸς Ἀστρονοµούµενος), i.e. works other than the ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
''. 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, 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 (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, Apollonius,
Aristaeus A minor god in Greek mythology, attested mainly by Athenian writers, Aristaeus (; ''Aristaios'' (Aristaîos); lit. “Most Excellent, Most Useful”), was the culture hero credited with the discovery of many useful arts, including bee-keeping; ...
and
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
, 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 ...
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 theor ...
, 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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
, and is followed by proofs that the conic is a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
,
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 (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, cal ...
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 Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch-born American (since 1934) mathematician, historian of mathematics and Marxian theoretician who spent most of his life in the U.S. Life Dirk Jan Struik was born in 1 ...
. 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 research university in Providence, Rhode Island. Brown is the seventh-oldest institution of higher education in the United States, founded in 1764 as the College in the English Colony of Rhode Island and Providenc ...
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 l ...
, used the relation of
projective harmonic conjugate In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line t ...
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 ''A'', ''B'', ''C'' and ''D'' on a line, th ...
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 it ...
is revealed as a lemma in Book VII.


Book VIII

Lastly, 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 virtually unknown to the Arabs and medieval Europeans, 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 humanist and mathematician. Born in Urbino, he studied at Padua and at Ferrara, where he received his doctorate in medicine. He was most famous for his central role as translator o ...
. Diophantus's ''Arithmetica'' 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 classical 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 engineerin ...
.
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 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 ...
also developed his version of analytic geometry and his method of Maxima and Minima from Pappus's summaries of Apollonius's lost works ''Plane Loci'' and ''On Determinate Section''. Other mathematicians influenced by Pappus were Pacioli,
da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
,
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 o ...
, 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: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
,
Bernoulli Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: ** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...
,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
,
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
,
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 becau ...
, 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: The ...
. AIP Conference Proceedings 1479, 9 (2012); https://doi.org/10.1063/1.4756049


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 ...
*
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, ''C''U and ''C''V, which are tangent at the point A a ...
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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 ...
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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 Greek ...


Notes


References

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Further reading

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External links


Pappos
(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 writers 290s births 350s deaths Year of birth unknown Year of death unknown 4th-century mathematicians