Pappus of Alexandria (/ˈpæpəs/ ⓘ;Ancient Greek:Πάππος ὁ Ἀλεξανδρεύς;c. 290 – c. 350 AD) was aGreek mathematician oflate antiquity known for hisSynagoge (Συναγωγή) orCollection (c. 340),[1] and forPappus's hexagon theorem inprojective 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 inAlexandria, where he worked as amathematics teacher to higher level students, one of whom was named Hermodorus.[2]
TheCollection, 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, includinggeometry,astronomy, andmechanics.[1]
Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to some, he stands out as a remarkable exception[3] and, to others, as an exemplar of ills that halted the progress of Greek science.[4] In many respects, his fate strikingly resembles that ofDiophantus', originally of limited importance but becoming very influential in the lateRenaissance andEarly Modern periods.
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 thanPtolemy (died c. 168 AD), whom he quotes, and earlier thanProclus (bornc. 411), who quotes him.[3]
The 10th centurySuda states that Pappus was of the same age asTheon of Alexandria, who was active in the reign of EmperorTheodosius I (372–395).[5] A different date is given by a marginal note to a late 10th-century manuscript[3] (a copy of a chronological table by the same Theon), which states, next to an entry on EmperorDiocletian (reigned 284–305), that "at that time wrote Pappus".[6]
However, a verifiable date comes from the dating of a solar eclipse mentioned by Pappus himself. In his commentary on theAlmagest he calculates "the place and time of conjunction which gave rise to the eclipse inTybi in 1068 afterNabonassar". This works out as 18 October 320, and so Pappus must have been active around 320.[2]
The great work of Pappus, in eight books and titledSynagoge orCollection, has not survived in complete form: the first book is lost, and the rest have suffered considerably. TheSuda enumerates other works of Pappus:Χωρογραφία οἰκουμενική (Chorographiaoikoumenike orDescription of the Inhabited World), a commentary on the thirteen books ofPtolemy'sAlmagest (of which the part on books 5 and 6 survives),Ποταμοὺς τοὺς ἐν Λιβύῃ (The Rivers in Libya), andὈνειροκριτικά (The Interpretation of Dreams).[5] Pappus himself mentions another commentary of his own on theἈνάλημμα (Analemma) ofDiodorus of Alexandria. Pappus also wrote commentaries onEuclid'sElements (of which fragments are preserved inProclus and theScholia, while that on the tenth Book has been found in an Arabic manuscript), and on Ptolemy'sἉρμονικά (Harmonika).[3]
Federico Commandino translated theCollection 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 historianPaul ver Eecke was the first to publish a translation of theCollection into a modern European language; his two-volume, French translation has the titlePappus d'Alexandrie. La Collection Mathématique. (Paris and Bruges, 1933).[7]
Pappus'sCollection 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 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 hisCollection "a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us".[3]
The surviving portions ofCollection can be summarized as follows.[8]
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.[3]
The whole of Book II (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition)[3] discusses a method of multiplication from an unnamed book byApollonius 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 to2×1054 and2×1038.[9]
Book III contains geometrical problems, plane and solid. It may be divided into five sections:[3]
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 asarbelos ("shoemakers knife") form the first division of the book; Pappus turns then to a consideration of certain properties ofArchimedes's spiral, theconchoid of Nicomedes (already mentioned in Book I as supplying a method of doubling the cube), and the curve discovered most probably byHippias of Elis about 420 BC, and known by the name, τετραγωνισμός, orquadratrix. 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 thetrisection 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.[11]
In Book V, after an interesting preface concerning regular polygons, and containing remarks upon thehexagonal 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 (followingZenodorus'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 ofPlato. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered byArchimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.[11]
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 theAlmagest. It accordingly comments on theSphaerica ofTheodosius, theMoving Sphere ofAutolycus, Theodosius's book onDay and Night, the treatise ofAristarchusOn the Size and Distances of the Sun and Moon, and Euclid'sOptics and Phaenomena.[11]
SinceMichel Chasles cited this book of Pappus in his history of geometric methods,[12] 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 ofEuclid,Apollonius,Aristaeus andEratosthenes, thirty-three books in all, the substance of which he intends to give, with the lemmas necessary for their elucidation. With the mention of thePorisms of Euclid we have an account of the relation ofporism 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 afterPaul Guldin, but appear to have been discovered by Pappus himself.[11]
Book VII also contains
Chasles's citation of Pappus was repeated byWilhelm Blaschke[14] andDirk Struik.[15] In Cambridge, England, John J. Milne gave readers the benefit of his reading of Pappus.[16] In 1985 Alexander Jones wrote his thesis atBrown 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 thecomplete quadrangle, used the relation ofprojective harmonic conjugates, and displayed an awareness ofcross-ratios of points and lines. Furthermore, the concept ofpole and polar is revealed as a lemma in Book VII.[17]
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 ofconjugate diameters are given.[11]
Pappus'sCollection was relatively unknown in medieval Europe, but exerted great influence on 17th-century mathematics after being translated to Latin byFederico Commandino.[18]Diophantus'sArithmetica and Pappus'sCollection were the two major sources ofViète'sIn artem analyticam isagoge (1591).[19] The Pappus's problem and its generalization ledDescartes to the development ofanalytic geometry.[20]Fermat also developed his version of analytic geometry and his method ofMaxima and Minima from Pappus's summaries ofApollonius's lost worksPlane Loci andOn Determinate Section.[21] Other mathematicians influenced by Pappus werePacioli,da Vinci,Kepler,van Roomen,Pascal,Newton,Bernoulli,Euler,Gauss,Gergonne,Steiner andPoncelet.[22]
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 areDescription of the Inhabited World; a commentary on the four books of theGreat Syntaxis of Ptolemy;The Rivers in Libya; andThe Interpretation of Dreams.
Attribution:
This article incorporates text from a publication now in thepublic domain: Heath, Thomas Little (1911). "Pappus of Alexandria". InChisholm, Hugh (ed.).Encyclopædia Britannica. Vol. 20 (11th ed.). Cambridge University Press. pp. 470–471.Greek mathematician, astronomer, and geographer whose chief importance lies in his commentaries on the mathematical work of his predecessors
