Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of antiquity, known for his Synagoge or Collection, and for Pappus's hexagon theorem in projective geometry. 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.
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, recreational mathematics, doubling the cube, polygons and polyhedra.
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 writes. "In this respect the fate of Pappus strikingly resembles that of Diophantus.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 at which he himself wrote. If no other date information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus, who quotes him.The Suda states that Pappus was of the same age as Theon of Alexandria, who was active in the reign of Emperor Theodosius I. A different date is given by a marginal note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian, that "at that time wrote Pappus".
However, a real date comes from the dating of a solar eclipse mentioned by Pappus himself, when in his commentary on the Almagest he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi in 1068 after Nabonassar". This works out as October 18, 320, and so Pappus must have been writing 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 enumerates other works of Pappus: Χωρογραφία οἰκουμενική, commentary on the four books of Ptolemy's Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, and Ὀνειροκριτικά. Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclid's Elements, and on Ptolemy's Ἁρμονικά.Federico Commandino translated the Collection of Pappus into Latin in 1588. The German classicist and mathematical historian Friedrich Hultsch published a definitive 3-volume presentation of Commandino's translation with both the Greek and Latin versions. 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 2-volume, French translation has the title Pappus d'Alexandrie. La Collection Mathématique.
''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 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 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 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 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 polytopes.
- An addition by a later writer on another solution of the first problem of the book.
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, 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.
According to the preface, Book VI is intended to resolve difficulties occurring in the so-called "Lesser Astronomical Works", i.e. works other than the Almagest. It accordingly comments on the Sphaerica of Theodosius, 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 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, Apollonius, 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 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 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; ; the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself.
Book VII contains also
- 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;
- 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.
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 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 conjugates, and displayed an awareness of cross-ratios of points and lines. Furthermore, the concept of pole and polar 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 are given.Theorems
Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.He also gives his name to the Pappus chain, and to the Pappus configuration and Pappus graph arising from his hexagon theorem.