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{{Short description|4th-century Greek mathematician (c. 290–350)}}
{{About|the Greek mathematician|the flower structure|Pappus (botany)}}
{{Use dmy dates|date=October 2022}}
[[File:Pappi Alexandrini. Mathematicae collectiones. 1588.jpg|thumb|Title page of Pappus's ''Mathematicae Collectiones'', translated into Latin by [[Federico Commandino]] (1588).]]

'''Pappus of Alexandria''' ({{IPAc-en|ˈ|p|æ|p|ə|s|audio=LL-Q1860 (eng)-Flame, not lame-Pappus.wav}}; {{langx|grc|Πάππος ὁ Ἀλεξανδρεύς}}; {{circa| 290| 350}} AD) was a [[Greek mathematics|Greek mathematician]] of [[late antiquity]] known for his ''Synagoge'' (Συναγωγή) or ''Collection'' ({{circa| 340}}),<ref name=":0">{{Cite book |last=Bird |first=John |url=https://books.google.com/books?id=bAcqDwAAQBAJ |title=Engineering Mathematics |date=14 July 2017 |publisher=Taylor & Francis |isbn=978-1-317-20260-8 |pages=590 |language=}}</ref> and for [[Pappus's hexagon theorem]] in [[projective geometry]]. 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]], where he worked as a [[Mathematics education|mathematics teacher]] to higher level students, one of whom was named Hermodorus.<ref name=dedron>Pierre Dedron, J. Itard (1959) ''Mathematics And Mathematicians'', Vol. 1, p. 149 (trans. [[Judith V. Field]]) (Transworld Student Library, 1974)</ref>

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]], [[astronomy]], and [[mechanics]].<ref name=":0" />

Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to some, he stands out as a remarkable exception{{sfn|Heath|1911|p=740}} and, to others, as an exemplar of ills that halted the progress of Greek science.<ref>[[William Whewell]], [https://www.gutenberg.org/cache/epub/68693/pg68693-images.html ''History of the Inductive Sciences''], Vol.1, pp.188-189 (New York: [[D. Appleton & Company]], 1875) (retrieved Mar. 22, 2025).</ref> In many respects, his fate strikingly resembles that of [[Diophantus|Diophantus']], originally of limited importance but becoming very influential in the late [[Renaissance]] and [[Scientific Revolution|Early Modern]] 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]] (died c. 168 AD), whom he quotes, and earlier than [[Proclus]] (born {{circa| 411}}), who quotes him.{{sfn|Heath|1911|p=740}}

The 10th century ''[[Suda]]'' states that Pappus was of the same age as [[Theon of Alexandria]], who was active in the reign of Emperor [[Theodosius I]] (372–395).<ref name=sol>

{{cite web |url=http://www.stoa.org/sol-bin//search.pl?search_method=QUERY&login=&enlogin=&searchstr=pi,265&field=adlerhw_gr&db=REAL |title=Suda On Line – Pappos |publisher=Suda On Line and the Stoa Consortium |access-date=11 July 2012 |editor-first=David |editor-last=Whitehead |quote=Alexandrian, philosopher, born in the time of the elder emperor Theodosius, when the philosopher Theon also flourished, the one who wrote about Ptolemy's Canon. His books are ''Description of the Inhabited World''; a commentary on the four books of the ''Great Syntaxis'' of Ptolemy; ''The Rivers in Libya''; and ''The Interpretation of Dreams''.}}</ref> A different date is given by a marginal note to a late 10th-century manuscript{{sfn|Heath|1911|p=740}} (a copy of a chronological table by the same Theon), which states, next to an entry on Emperor [[Diocletian]] (reigned 284–305), that "at that time wrote Pappus".<ref>{{cite thesis|title=Pappus Reborn: Pappus of Alexandria and the Changing Face of Analysis and Synthesis in Late Antiquity|year=2008|first=Bronwyn|last=Rideout|page=14|type=Master's thesis|publisher=University of Canterbury Humanities|hdl=10092/2329|doi=10.26021/3834}}</ref>

However, a verifiable date comes from the dating of a solar eclipse mentioned by Pappus himself.  In his commentary on the ''[[Almagest]]'' he calculates "the place and time of conjunction which gave rise to the eclipse in [[Month of Tobi|Tybi]] in 1068 after [[Nabonassar]]". This works out as 18 October 320, and so Pappus must have been active around 320.<ref name=dedron />

==Works==
[[File:Pappus - Mathematicae collectiones, 1660.jpg|thumb|''Mathematicae collectiones'', 1660]]

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]]'' enumerates other works of Pappus: ''Χωρογραφία οἰκουμενική'' (''[[Chorography|Chorographia]] [[ecumene|oikoumenike]]'' or ''Description of the Inhabited World''), a commentary on the thirteen books of [[Ptolemy]]'s ''[[Almagest]]'' (of which the part on books 5 and 6 survives), ''Ποταμοὺς τοὺς ἐν Λιβύῃ'' (''The Rivers in Libya''), and ''Ὀνειροκριτικά'' (''The Interpretation of Dreams'').<ref name=sol /> Pappus himself mentions another commentary of his own on the ''Ἀνάλημμα'' (''[[Analemma]]'') of [[Diodorus of Alexandria]]. Pappus also wrote commentaries on [[Euclid]]'s [[Euclid's Elements|''Elements'']] (of which fragments are preserved in [[Proclus]] and the [[Scholia]], while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy's ''Ἁρμονικά'' (''Harmonika'').{{sfn|Heath|1911|p=740}}

[[Federico Commandino]] 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).<ref>{{cite journal|author=Smith, David Eugene|author-link=David Eugene Smith|title=Review of ''Pappus d'Alexandrie. La Collection Mathématique.'' by Paul ver Eecke|journal=Bull. Am. Math. Soc.|date=January 1934|volume=40|issue=1|pages=11–12|doi=10.1090/S0002-9904-1934-05766-5|url=https://www.ams.org/journals/bull/1934-40-01/S0002-9904-1934-05766-5/S0002-9904-1934-05766-5.pdf|doi-access=free}}</ref>

==''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. [[Thomas Heath (classicist)|Heath]] 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".{{sfn|Heath|1911|p=740}}

The surviving portions of ''Collection'' can be summarized as follows.<ref>{{cite journal|title=Pappus. introductory paper|author=Weaver, James Henry|journal=Bull. Amer. Math. Soc.|volume=23|year=1916|issue=3|pages=127–135|doi=10.1090/S0002-9904-1916-02895-3|doi-access=free}}</ref>

[[File:Pappi Alexandrini, Mathematicae Collectiones, 1589 02.jpg|thumb|Pages from ''Mathematicae Collectiones'', published in Venice in 1589.]]

===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.{{sfn|Heath|1911|p=740}}

===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){{sfn|Heath|1911|p=740}} discusses a method of multiplication from an unnamed book by [[Apollonius of Perga]]. 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 {{val|2e54}} and {{val|2e38}}.<ref>Pappus of Alexandria, trans. into Latin by Friedrich Hultsch. [https://archive.org/details/pappialexandrin01hultgoog ''Pappi Alexandrini collectionis quae supersunt'']. Apud Weidmannos, 1877, pp.&nbsp;19–29.</ref>

===Book III===
Book III contains geometrical problems, plane and solid. It may be divided into five sections:{{sfn|Heath|1911|p=740}}

# 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]] 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.{{sfn|Heath|1911|p=740}}
# 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.{{sfn|Heath|1911|p=740}}
# On a curious problem suggested by Euclid I. 21.{{sfn|Heath|1911|p=740}}
# On the inscribing of each of the five regular polyhedra in a sphere.{{sfn|Heath|1911|p=740}} Here Pappus observed that a [[regular dodecahedron]] and a [[regular icosahedron]] 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 polytope|dual polytopes]].<ref name="Coxeter2012">{{cite book|author=H. S. M. Coxeter|title=Regular Polytopes|url=https://books.google.com/books?id=2ee7AQAAQBAJ&pg=PP1|date=23 May 2012|publisher=Courier Corporation|isbn=978-0-486-14158-9|page=88 238}}</ref>
# An addition by a later writer on another solution of the first problem of the book.{{sfn|Heath|1911|p=740}}

===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]]), 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]] ("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]] about 420 BC, and known by the name, τετραγωνισμός, or [[quadratrix]]. 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]], 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.{{sfn|Heath|1911|p=741}}

===Book V===
In Book V, after an interesting preface concerning regular polygons, and containing remarks upon the [[Honeycomb conjecture|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 (mathematician)|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]]. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by [[Archimedes]], and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.{{sfn|Heath|1911|p=741}}

===Book VI===
According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "[[Little Astronomy]]" (Μικρὸς Ἀστρονομούμενος), i.e. works other than the ''[[Almagest]]''. It accordingly comments on the ''Sphaerica'' of [[Theodosius of Bithynia|Theodosius]], the ''Moving Sphere'' of [[Autolycus of Pitane|Autolycus]], Theodosius's book on ''Day and Night'', the treatise of [[Aristarchus of Samos|Aristarchus]] ''[[On the Sizes and Distances (Aristarchus)|On the Size and Distances of the Sun and Moon]]'', and Euclid's ''Optics and Phaenomena''.{{sfn|Heath|1911|p=741}}

===Book VII===
Since [[Michel Chasles]] cited this book of Pappus in his history of geometric methods,<ref>Michel Chasles (1837) ''Aperçu historique sur l'origine et le développement des méthodes en géométrie'', especially page 302; see also pages 12, 78, and 518.</ref> 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]], [[Apollonius of Perga|Apollonius]], [[Aristaeus the Elder|Aristaeus]] and [[Eratosthenes]], 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]] 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]], but appear to have been discovered by Pappus himself.{{sfn|Heath|1911|p=741}}

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;{{sfn|Heath|1911|p=741}}
# important lemmas on the ''Porisms'' of Euclid,{{sfn|Heath|1911|p=741}} including what is called [[Pappus's hexagon theorem]];{{sfn|Heath|1911b|ref=Heath1911b|p=102}}
# 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]], and is followed by proofs that the conic is a [[parabola]], [[ellipse]], or [[hyperbola]] 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).{{sfn|Heath|1911|p=741}}

Chasles's citation of Pappus was repeated by [[Wilhelm Blaschke]]<ref>Wilhelm Blaschke (1948) ''Projektiva Geometrie'', page 140</ref> and [[Dirk Struik]].<ref>Dirk Struik (1953) ''Lectures in Analytic and Projective Geometry'', page 19, Addison-Wesley</ref> In Cambridge, England, John J. Milne gave readers the benefit of his reading of Pappus.{{sfn|Milne|1911}} In 1985 Alexander Jones wrote his thesis at [[Brown University]] 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]], used the relation of [[projective harmonic conjugate]]s, and displayed an awareness of [[cross-ratio]]s of points and lines. Furthermore, the concept of [[pole and polar]] is revealed as a lemma in Book VII.<ref>{{harvtxt|Jones|1986}}. For the complete quadrangle, cross-ratios, and harmonic conjugates, see, e.g., p. 560. For discussion of Pappus's results on poles and polars, see, e.g., p. 568.</ref>

===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]] are given.{{sfn|Heath|1911|p=741}}

==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]].<ref>{{cite journal |last1=Marchisotto |first1=Elena Anne|author-link=Elena Marchisotto |title=The Theorem of Pappus: A Bridge between Algebra and Geometry |journal=The American Mathematical Monthly |date=June 2002 |volume=109 |issue=6 |pages=497–516 |doi=10.2307/2695440 |jstor=2695440 }}</ref> [[Diophantus]]'s ''[[Arithmetica]]'' and Pappus's ''Collection'' were the two major sources of [[Viète]]'s ''In artem analyticam isagoge'' (1591).<ref>{{cite journal |last1=Forbes |first1=Eric G |title=Descartes and the birth of analytic geometry |journal=Historia Mathematica |date=May 1977 |volume=4 |issue=2 |pages=141–151 |doi=10.1016/0315-0860(77)90105-7 |doi-access=free }}</ref> The Pappus's problem and its generalization led [[Descartes]] to the development of [[analytic geometry]].<ref>{{Cite journal|url=https://www.scientificamerican.com/article/the-invention-of-analytic-geometry/|doi=10.1038/scientificamerican0149-40|title=The Invention of Analytic Geometry|year=1949|last1=Boyer|first1=Carl B.|journal=Scientific American|volume=180|issue=1|pages=40–45|bibcode=1949SciAm.180a..40B}}</ref> [[Fermat]] also developed his version of analytic geometry and his method of [[Fermat's theorem (stationary points)|Maxima and Minima]] from Pappus's summaries of [[Apollonius of Perga|Apollonius]]'s lost works ''Plane Loci'' and ''On Determinate Section''.<ref>{{cite journal |last1=Mahoney |first1=Michael S. |title=Fermat's Mathematics: Proofs and Conjectures: Fermat's working habits as a mathematician shed new light on the mystery of his famous 'last theorem' |journal=Science |date=6 October 1972 |volume=178 |issue=4056 |pages=30–36 |doi=10.1126/science.178.4056.30 |pmid=17754730 |jstor=1734005 }}</ref> Other mathematicians influenced by Pappus were [[Pacioli]], [[da Vinci]], [[Kepler]], [[Adriaan van Roomen|van Roomen]], [[Blaise Pascal|Pascal]],  [[Isaac Newton|Newton]], [[Jacob Bernoulli|Bernoulli]], [[Euler]], [[Gauss]], [[Gergonne]], [[Jakob Steiner|Steiner]] and [[Jean-Victor Poncelet|Poncelet]].<ref>{{cite journal |last1=Wanner |first1=Gerhard |title=The importance of Pappus for the development of mathematics |journal=Numerical Analysis and Applied Mathematics Icnaam 2012: International Conference of Numerical Analysis and Applied Mathematics |series=AIP Conference Proceedings |date=2012 |volume=1479 |issue=1 |pages=9–10 |doi=10.1063/1.4756049 |bibcode=2012AIPC.1479....9W }}</ref>

==See also==
*[[Pappus's hexagon theorem]]
*[[Pappus's centroid theorem]]
*[[Pappus chain]]
*[[Pappus configuration]] 
*[[Pappus graph]]

== Notes ==
{{reflist}}

==References==
* {{cite EB1911 |last=Heath |first=Thomas Little |wstitle=Porism |volume=22 |pages=102–103|ref=Heath1911b}}
* {{cite book
 | editor-first = Alexander | editor-last = Jones
 | doi = 10.1007/978-1-4612-4908-5
 | isbn = 9781461249085
 | location = New York
 | publisher = Springer
 | series = Sources in the History of Mathematics and Physical Sciences
 | title = Pappus of Alexandria Book 7 of the Collection
 | year = 1986}}
* {{cite book |last=Milne |first=John J. |year=1911 |url=https://archive.org/details/elementarytreati00milnuoft |title= An Elementary Treatise on Cross-Ratio Geometry with Historical Notes |publisher=Cambridge University Press |page=[https://archive.org/details/elementarytreati00milnuoft/page/11 11]}} <!--quote=Milne Cross-ratio (page 11). -->

Attribution:
* {{EB1911|last=Heath| |first=Thomas Little |wstitle=Pappus of Alexandria|volume=20 |pages=470–471}}

==Further reading==
* {{cite encyclopedia|ref=none |last=Jones |first=Alexander Raymond  |date=19 January 2017 |encyclopaedia=Encyclopædia Britannica |title=Pappus of Alexandria |url=http://www.britannica.com/EBchecked/topic/442170/Pappus-of-Alexandria }}

* {{cite encyclopedia|ref=none   |encyclopedia=The Hutchinson Dictionary of Scientific Biography |publisher=Helicon Publishing |year=2004 |title=Pappus of Alexandria (lived c. AD 200–350)| quote=Greek mathematician, astronomer, and geographer whose chief importance lies in his commentaries on the mathematical work of his predecessors}}

* {{cite book|ref=none  |first=Paul Ver |last=Eecke |year=1933 |title=Pappus d'Alexandrie: La Collection Mathématique avec une Introduction et des Notes |edition=2 volumes Fondation Universitaire de Belgique |location=Paris |publisher=Albert Blanchard}}

==External links==
{{wikiquote}}

* {{cite book |last1=Pappi |title=Mathematicae collectiones |date=1660 |publisher=ex typographia HH. de Duccijs |url=https://books.google.com/books?id=6ZBoAAAAcAAJ |language=la}}
* {{cite book |title=Pappi Alexandrini Mathematicae collectiones. A Federico Commandino Vrbinatae in latinum conuersae, & commentarijs illustratae |date=1589 |publisher=apud Franciscum de Franciscis Senensem |url=https://archive.org/details/bub_gb_YTKUNyiY8sEC/page/n17/mode/2up |language=Russian}}
* [https://wilbourhall.org/index.html#pappus Scans of Hultsch's Edition of Pappus of Alexandria's Collection] at wilbourhall.org
* [http://www.hs-augsburg.de/~harsch/graeca/Chronologia/S_post04/Pappos/pap_intr.html Pappos] (Bibliotheca Augustana)
* {{MacTutor Biography|id=Pappus|mode=cs1}}
* [http://www.answers.com/topic/pappus-1 "Pappus"], ''Columbia Electronic Encyclopedia'', Sixth Edition at Answer.com.
* [http://www.mathpages.com/home/kmath542/kmath542.htm Pappus's Theorem] at MathPages
* [https://web.archive.org/web/20070715043525/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1186&bodyId=1319 Pappus's work on the Isoperimetric Problem] at [http://mathdl.maa.org/convergence/1/ Convergence]

{{Greek mathematics}}
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[[Category:Roman-era Alexandrians]]
[[Category:Ancient Greek geometers]]
[[Category:4th-century Greek writers]]
[[Category:290s births]]
[[Category:350s deaths]]
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[[Category:4th-century mathematicians]]