Pope of Alexandria

Cover of the Mathematicae Collections by Federico Commandino (1589).

Papus of Alexandria (as eponymous Pappus, in Greek Πάππος ὁ Ἀλεξανδρεύς) (c. 290 – c. 350) was one of the last great Greek mathematicians of the 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 wide variety of problems in geometry, recreational mathematics, doubling the cube, polygons and polyhedra.

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

"On top was his contemporaries, his work was so little understood, that there are no references to him in other Greek writers; and therefore his work had no effect on stopping the decay of mathematics. In this regard, the fate of Papo is strikingly like that of Diofanto.

The Synagoge was translated into Latin in 1588 by Federico Commandino. The historian of mathematics and classicist Friedrich Hultsch published in 1878 the definitive version in Greek and Latin of Papus. Paul Ver Eecke, a 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's Theorem"). Among these are:

• Theorem of the center of Papo,
• Theorem harmonic of Papo,
• Theorem of the hexagon of Papo.

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 the Papo chain.

Works

Mathematicae collectiones, 1660

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

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 the versions Greek and Latin (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; the translation of it into French in 2 volumes is titled Pappus d'Alexandrie. La Collection Mathématique (Paris and Bruges, 1933).

Collection

The characteristics of the Papo Collection are that it contains a list, systematically ordered, of the most important results obtained by his predecessors and, secondly, explanatory or expansion notes on previous discoveries. These discoveries form, in fact, a text on which Papo expands discursively. Heath found the systematic introductions to the various books valuable, as they clearly set out an outline of the content and general scope of the topics to be covered. From these introductions one can judge the style of Papo's writing, which is excellent and even elegant the moment it is released from the shackles of mathematical formulas and expressions. Heath also found that his characteristic accuracy made his Collection & # 34; a most admirable substitute for the texts of the many valuable earlier mathematical treatises of which time has deprived us & # 3. 4;.

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, Book III clearly introducing itself as the beginning of a new subject.

The entire Book II (the first part of which is lost, the extant fragment beginning in the middle of proposition 14) discusses a multiplication method from an unnamed book by Apollonius of Perga. The final propositions try to multiply together the numerical values of the Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2x10⁵⁴ and 2x10³⁸.

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

1. On the famous problem of finding two proportional averages between two given lines, which emerged from the double cube, reduced by Hippocrates de Quíos to the first. Papo gives several solutions to this problem, including a method of making successive approaches to the solution, whose importance apparently did not appreciate; he adds his own solution to the more general problem of geometrically finding the side of a cube whose content is in any proportion to that of a given one.
2. On arithmetic, geometric and harmonic means between two straights, and the problem of representing the three in the same geometric figure. This serves as an introduction to a general media theory, of which Papo distinguishes ten types, and gives a table that represents examples of each in whole numbers.
3. About a curious problem suggested by Euclides I. 21.
4. On the registration of each of the five regular polyhedrons in one sphere. Here Papo observed that a regular dodecahedron and a regular icosahedron could register in the same sphere so that their vertices were all in the same 4 circles of latitude, with 3 of the 12 vertices of icosahedron in each circle, and 5 of the 20 vertices of dodecahedron in each circle. This observation has spread to dual polytopes of greater dimension.
5. An addition of a later writer on another solution of the first problem of the book.

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 there is the well-known generalization of Euclid I.47 (Papo's area theorem), then several theorems about the circle follow, which lead to the problem of constructing a circle circumscribing three given circles, that touches two and two. This and several other propositions about contact, for example, the cases of circles touching each other and being inscribed in the figure made of three semicircles and known as arbelos ("shoemaker's knife") form the first division of the book; Papo then goes on to consider certain properties of the Archimedean spiral, the Nicomedean conchoid (already mentioned in Book I as a method of doubling the cube), and the curve most likely discovered by Hippias of Elis around 420 BC. C., and known by the name of τετραγωνισμός, or quadratrix. Proposition 30 describes the construction of a double curvature curve 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 rotates uniformly about its diameter, the point describing one quadrant and the great circle one 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 squaring a curved surface. The rest 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 squaring and spiraling. In one of the solutions to the first problem is the first recorded use of the property of a conic (a hyperbola) with reference to focus and directrix.

In Book V, after an interesting preface on regular polygons, which contains observations on the hexagonal shape of the cells of the honeycombs, Papo compares the areas of different plane figures that all have the same perimeter (following Zenodorus's treatise on this subject), and of the volumes of different solid figures that all have the same surface area, and, finally, a comparison of the five regular solids of Plato. Incidentally, Papo describes the other thirteen polyhedra bounded by equilateral and equiangular, but not similar, polygons discovered by Archimedes, and finds, by a method reminiscent of Archimedeans, the surface area and volume of a sphere.

According to the preface, Book VI is intended to resolve the difficulties that arise in the so-called Minor Astronomical Works (Μικρὸς Ἀστρονοµούµενος), that is, the different works of the Almagest. Consequently, he comments on Theodosius' Sphaerica, Autolycus' Moving Sphere, Theodosius's book on Day and Night, the treatise on Aristarchus On the size and distances of the Sun and the Moon, and Euclid's Optics and Phenomena.

Book VII

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

The preface to Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem. Next, Papo lists the works of Euclid, Apollonius, Aristeo and Eratosthenes, thirty-three books in all, the substance of which he claims to give, with the necessary mottos for their elucidation. With the mention of Euclid's Porismos we have an account of the relationship of porismo with the theorem and the problem. Included in the same preface is (a) the famous problem known by the name of Papo, often stated thus: Given a series of 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 given inclinations a, the given lines satisfy the condition that the product of some of them can bear a constant relationship with the product of the rest; (Papo does not express it in this way, but by means of the composition of ratios, saying that if the ratio is given that is composed of the ratios of the pairs, one of a set and one of another of the lines thus drawn, and of the ratio of the odd, if any, to a given line, the point will be on a given curve in position); (b) theorems that were rediscovered by Paul Guldin and named after him, but which appear to have been discovered by Papo himself.

Book VII also contains

1. under the title of De Sectione Determinata of Apollonius, slogans that, closely examined, are seen as cases of the involution of six points;
2. important slogans on Porisms of Euclides, including the call Papo hexagon theorem;
3. a motto over Surface loci of Euclides who affirms that the place of a point such that its distance to a given point is a constant relationship with its distance to a given straight is a conical, and is followed by proof that the conical is a parable, an ellipse or a hyperbola according to the constant relationship is equal, less or greater than 1 (the first recorded tests of the properties, which do not appear in Apollonium).

Chasles de Papo's quote was repeated by Wilhelm Blaschke and Dirk Struik. 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 manages to show how Papo manipulated the complete quadrilateral, used the relation of projective harmonic conjugates, and showed an awareness of the cross relation of points and lines. Also, the concept of pole and polar is revealed as a motto in Book VII.

Book VIII

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