## Summary

Pappus of Alexandria (as eponym Pappus, Greek Πάππος ὁ Ἀλεξανδρεύς) (c. 290 – c. 350) was one of the last great Greek mathematicians of Antiquity, known for his work Synagoge (c. 340). Hardly anything is known of his life, except that he was a teacher in Alexandria and that he had a brother named Hermodorus.

His Synagoge (Collection) is his best known work. It is an eight-volume compendium of mathematics. It deals with a great variety of geometry problems, recreational mathematics, cube duplication, polygons and polyhedra.

Papo lived in the first half of the fourth century. His figure stands out from the general stagnation of mathematics of his time.

The Synagoge was translated into Latin in 1588 by Federico Commandino. The historian of mathematics and classicist Friedrich Hultsch (1833-1908) published the definitive Greek and Latin version of Papus in 1878. Paul Ver Eecke, Belgian historian, translated the work into French in 1933.

In geometry, several theorems are attributed to him, all known by the generic name of “Papo’s Theorem” (or “Pappus’ Theorem”). Among these are:

He also investigated a geometric figure consisting of a ring of circles drawn between two circles tangent to each other. This figure is known as Papo’s chain.

Pappus’ great work, in eight books and entitled Synagoge or Collection, has not survived complete: the first book has been lost, and the rest have suffered quite a bit. The Suda lists other works of Pappus: Χωρογραφία οἰκουμενική (Chorographia oikoumenike or Description of the Inhabited World), a commentary on the four books of Ptolemy’s Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ’ (The Rivers of Libya), and Ὀνειροκριτικά (The Interpretation of Dreams). Pappus himself mentions another commentary of his on the Ἀνάλημμα (Analemma) of Diodorus of Alexandria. Pappus also wrote commentaries on Euclid’s Elements in Elements’ (fragments of which are preserved in Proclus and the Scholia, while that of the tenth Book has been found in an Arabic manuscript), and on Ptolemy’s Ἁρρμονικά (Harmonika).

Federico Commandino translated the Pappus Collection 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 the Greek and Latin versions (Berlin, 1875-1878). Building on 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 is entitled Pappus d’Alexandrie. La Collection Mathématique (Paris and Bruges, 1933).

The characteristics of Papo’s Collection are that it contains a list, systematically ordered, of the most important results obtained by his predecessors and, secondly, explanatory notes or notes of amplification of the previous discoveries. These discoveries form, in fact, a text on which Papo discursively expands. Heath considered the systematic introductions to the various books valuable, since they clearly set out an outline of the contents and general scope of the subjects to be treated. From these introductions one can judge the style of Papo’s writing, which is excellent and even elegant the moment he frees himself from the shackles of mathematical formulas and expressions. Heath also found that his characteristic accuracy 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 parts of the Collection can be summarized as follows.

We can only conjecture that the lost Book I, like Book II, dealt with arithmetic, with Book III being clearly introduced as the beginning of a new subject.

The whole of Book II (the first part of which has been lost, the extant fragment beginning in the middle of proposition 14) discusses a method of multiplication from an unnamed book by Apollonius of Perga. The final propositions attempt to multiply together the numerical values of the Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2×1054 and 2×1038.

Book III contains geometric, plane and solid problems. It can be divided into five sections.

From Book IV the title and the preface have been lost, so the program has to be taken from the book itself. At the beginning is the well-known generalization of Euclid I.47 (Papo’s area theorem), then follow several theorems on the circle, leading to the problem of constructing a circle circumscribing three given circles, which touch two and two. This and several other propositions on contact, for example, the cases of circles touching each other and inscribed in the figure made of three semicircles and known as arbels (Papo then goes on to consider certain properties of the spiral of Archimedes, the conchoid of Nicomedes (already mentioned in Book I as a method for doubling the cube), and the curve most probably discovered by Hipias of Elis about 420 B.C., and known by the name of τετραγωνισμός, or quadrature. Proposition 30 describes the construction of a curve of double curvature called by Papo the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which in turn revolves 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 case of quadrature of a curved surface. The remainder of the book deals with the trisection of an angle, and with the solution of more general problems of the same type by means of quadrature and the spiral. In one of the solutions of the first problem is found 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 on the hexagonal form of honeycomb cells, Papo devotes himself to the comparison of the areas of different plane figures which all have the same perimeter (following Zenodorus’ treatise on this subject), and of the volumes of different solid figures which all have the same surface area, and, finally, a comparison of Plato’s five regular solids. Incidentally, Papo describes the other thirteen polyhedra limited by equilateral and equiangular polygons, but not similar, discovered by Archimedes, and finds, by a method reminiscent of Archimedes’, the surface area and volume of a sphere.

According to the preface, Book VI is intended to resolve the difficulties encountered in the so-called Minor Astronomical Works (Μικρὸς Ἀστρονοµούµενος), i.e., works other than the Almagest. Accordingly, he comments on Theodosius’ Sphaerica, Autolytus’ Moving Sphere, Theodosius’ book On Day and Night, Aristarchus’ treatise On the Size and Distances of the Sun and Moon, and Euclid’s Optics and Phenomena.

## Book VII

Since Michel Chasles cited this book by Papo in his history of geometrical methods, it has become the subject of considerable attention.

The preface to Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem. Papo then enumerates the works of Euclid, Apollonius, Aristeus, 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 Euclid’s Porisms we have an account of the relation of the porism to the theorem and the problem. In the same preface is included (a) the famous problem known by the name of Papo, often stated thus: Given a series of straight lines, find the locus of a point such that the lengths of the perpendiculars to, or (more generally) the lines drawn from it obliquely with inclinations given a, the given lines satisfy the condition that the product of some of them may bear a constant relation to the product of the remaining ones; (Papo does not express it in this form, but by means of the composition of ratios, saying that if the ratio is given which is composed of the ratios of the pairs one of a set and one of another of the lines thus traced, and of the ratio of the odd, if any, to a given straight line, the point will be on a given curve in position); (b) the theorems which were rediscovered by Paul Guldin and named later, but which seem to have been discovered by Papo himself.

Book VII also contains

Chasles’ quotation of Papo was repeated by Wilhelm Blaschke In Cambridge, England, John J. Milne gave readers the benefit of his reading of Papo. 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 Papo manipulated the complete quadrilateral, used the relation of projective harmonic conjugates, and showed an awareness of cross-relation of points and lines. In addition, the concept of pole and polar is revealed as a lemma in Book VII.

## Book VIII

Finally, Book VIII deals mainly with mechanics, the properties of the center of gravity and some mechanical powers. Some propositions on pure geometry are interspersed. Proposition 14 shows how to draw an ellipse through five given points, and proposition 15 gives a simple construction for the axes of an ellipse when a pair of conjugate diameters is given.

**Sources**

- Papo de Alejandría
- Pappus of Alexandria
- ^ a b Bird, John (14 July 2017). Engineering Mathematics. Taylor & Francis. p. 590. ISBN 978-1-317-20260-8.
- ^ a b Pierre Dedron, J. Itard (1959) Mathematics And Mathematicians, Vol. 1, p. 149 (trans. Judith V. Field) (Transworld Student Library, 1974)
- a b Heath, Thomas Little (1910-1911). «Encyclopædia Britannica». En Chisholm, Hugh, ed. Encyclopædia Britannica. A Dictionary of Arts, Sciences, Literature, and General information (en inglés) (11.ª edición). Encyclopædia Britannica, Inc.; actualmente en dominio público.
- Whitehead, David (ed.). “Suda On Line – Pappos”. Suda On Line y el Consorcio Stoa. Recuperado el 11 de julio de 2012. Alejandrino, filósofo, nacido en tiempos del emperador mayor Teodosio, cuando también floreció el filósofo Teón, el que escribió sobre el Canon de Ptolomeo. Sus libros son Descripción del mundo habitado; un comentario sobre los cuatro libros de la Gran Sintaxis de Ptolomeo; Los ríos de Libia; y La interpretación de los sueños.
- 1 2 Pappus Alexandrinus // Catalogue of the Library of the Pontifical University of Saint Thomas Aquinas
- Identifiants et Référentiels (фр.) — ABES, 2011.
- 1 2 3 4 Боголюбов, 1983, с. 363.
- En grec ancien Συναγωγή (traduit en français sous le titre de Collection mathématique).
- Viète, Fermat, Wallis, Simson, etc. Cf. à ce sujet ver Eecke 1933, Introduction.