Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat ( Oeuvres, i., 1891, pp. 3 – 51 ) and F. Schooten ( Leiden, 1656 ) but also, most successfully of all, by R. Simson ( Glasgow, 1749 ).

In his extant writings, Pappus gives no indication of the date of the authors whose treatises he makes use of, or of the time ( but see below ) at which he himself wrote.

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.

He also gives his name to the Pappus chain, and to the Pappus configuration and Pappus graph arising from his hexagon theorem.

For as much as we know of this lost treatise we are indebted to the Collection of Pappus of Alexandria, who mentions it along with other geometrical treatises, and gives a number of lemmas necessary for understanding it.

Pappus gives a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case.

Pappus gives also a complete enunciation of one porism of the first book of Euclid's treatise.

The rest of the enunciations given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms ; and these include 171 theorems.

The lemmas which Pappus gives in connexion with the porisms are interesting historically, because he gives:

About the same time Pierre de Fermat wrote a short work under the title Porismatum euclidaeorum renovata doctrina et sub forma isagoges recentioribus geometeis exhibita ( see Œuvres de Fermat, i., Paris, 1891 ); but two at least of the five examples of porisms which he gives do not fall within the classes indicated by Pappus.

This was followed in the same journal and in La Science by a controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of the text of Pappus, and declared himself in favour of the idea of Schooten, put forward in his Mathematicae exercitationes ( 1657 ), in which he gives the name of " porism " to one section.

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 B. C., and known by the name, τετραγωνισμός, or quadratrix.

In addition, through several of his works, most notably Philosophy from Oracles and Against the Christians, he was involved in a controversy with a number of early Christians, and his commentary on Euclid's Elements was used as a source by Pappus of Alexandria.

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.

262 190 BC ) posed and solved this famous problem in his work (, " Tangencies "); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived.

According to the 4th-century report of Pappus of Alexandria, Apollonius ' own book on this problem — entitled (, " Tangencies "; Latin: De tactionibus, De contactibus )— followed a similar progressive approach.

Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius – Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte – Coxeter graph, the Dyck graph, the Foster graph and the Biggs-Smith graph.

Barozzi translated many works of the ancients, including Proclus ’ s edition of Euclid's Elements ( published in Venice in 1560 ), as well as mathematical works by Hero, Pappus of Alexandria, and Archimedes.

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.

Again, Chasles seems to have been wrong in making the ten cases of the four-line Porism begin the book, instead of the intercept-Porism fully enunciated by Pappus, to which the " lemma to the first Porism " relates intelligibly, being a particular case of it.

A close friendship developed between Simson and Stewart, in part because of their mutual admiration of Pappus of Alexandria, which resulted in many curious communications with respect to the De Locis Planis of Apollonius of Perga and the Porisms of Euclid over the years.

A different date is given by a marginal note to a late 10th century manuscript ( a copy of a chronological table by the same Theon ), which states, next to an entry on Emperor Diocletian ( reigned 284 – 305 AD ), 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 ".

Pappus also wrote commentaries on Euclid's 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 ).

These discoveries form, in fact, a text upon which Pappus enlarges discursively.

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.

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 ( following 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.

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.

He first succeeded in explaining the only three propositions which Pappus indicates with any completeness.

xx., July, 1855 ), P. Breton published Recherches nouvelles sur les porismes d ' Euclide, in which he gave a new translation of the text of Pappus, and sought to base thereon a view of the nature of a porism more closely conforming to the definitions in Pappus.

The three porisms stated by Diophantus in his Arithmetica are propositions in the theory of numbers which can all be enunciated in the form " we can find numbers satisfying such and such conditions "; they are sufficiently analogous therefore to the geometrical porism as defined in Pappus and Proclus.

In 1986, Alexander Jones published a commentary on Book 7 of the Collection of Pappus of Alexandria, which Chasles had referred to in his history of geometric methods.

If ( x, y ) is a point on the parabola then, by Pappus ' definition of a parabola, it is the same distance from the directrix as the focus ; in other words:

Pappus &# 39 ; s theorem may refer to:

Each triangle can be extraverted in three different ways ; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the Pappus graph.

It is named after Pappus of Alexandria ; Pappus's hexagon theorem states that every two triples of collinear points ABC and abc ( none of which lie on the intersection of the two lines ) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points,, and.

Pappus himself mentions another commentary of his own on the Ἀνάλημμα ( Analemma ) of Diodorus of Alexandria.

# an arithmetical work ( see Pappus ) on a system both for expressing large numbers in language more everyday than that of Archimedes ' The Sand Reckoner and for multiplying these large numbers

* The Greek mathematician Pappus demonstrates geometrically the property of the center of gravity.

He continued his studies in Strasbourg, under the professor of Hebrew, Johannes Pappus ( 1549 – 1610 ), a zealous Lutheran, the crown of whose life's work was the forcible suppression of Calvinistic preaching and worship in the day, and who had great influence over him.

* The Hutchinson Dictionary of Scientific Biography ( Helicon Publishing, 2004 ) " Pappus of Alexandria ( lived c. AD 200-350 ) Greek mathematician, astronomer, and geographer whose chief importance lies in his commentaries on the mathematical work of his predecessors.