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Pappus of Alexandria

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. Collectionhis best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives.

It covers a wide range of topics, including geometryrecreational mathematicsdoubling the cubepolygons and polyhedra. Pappus was active in the 4th century AD. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. 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 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 died c.

The 10th century Suda states that Pappus was of the same age as Theon of Alexandriawho was active in the reign of Emperor Theodosius I — 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 after Nabonassar ".

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This works out as 18 Octoberand so Pappus must have been writing around The great work of Pappus, in eight books and titled Synagoge or Collectionhas not survived in complete form: the first book is lost, and the rest have suffered considerably.

Federico Commandino translated the Collection of Pappus into Latin in 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 Berlin, — 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.

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.

The whole of Book II the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition [2] discusses a method of multiplication from an unnamed book by Apollonius of Perga.

Book III contains geometrical problems, plane and solid.

Pappus of Alexandria Jordan, Alan, Justin, and Armia for Mrs. Herren's class

It may be divided into five sections: [2]. 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. This and several other propositions on contact, e.

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.

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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 the trisection of an angleand 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. In Book Vafter an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombsPappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter following Zenodorus 's treatise on this subjectand 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 Archimedesand finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

Since Michel Chasles cited this book of Pappus in his history of geometric methods, [9] 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 of Alexandria -was a Hellenized Egyptian born in Alexandria.

Very little is known about his life, but the written records suggest he was probably a teacher. His main contribution to mathematics was primarily as an encyclopedist. Pappus summarized and enlarged all of Greek mathematics in his work Synagoge The Mathematical Collection of which eight volumes survived perhaps originally in twelve of which the first and part of the second are missing.

The Collection is sometimes the only source for our knowledge of his predecessors' achievements. For instance, in his commentary to Almagest he determined the eclipse in AD. For his own originality, even if his chief importance is as the preserver of Greek scientific knowledge, Pappus stands with Diophantus as the last of the long and distinguished line of Alexandrian mathematicians Hutchinson dictionary of scientific biography. The Collection contains supplements to earlier treatises including geometry, recreational mathematics, doubling the cube, polygons and polyhedra, astronomy, and mechanics.

It dates from the late third century A. Pappus not only reproduces known solutions to geometric problems, but he frequently gives own solutions, or improvements and extensions to existing solutions and theorem.

For instance, Pappus handles the problem of inscribing five regular solids in a sphere in a way quite different from Euclid.

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Gives a generalization to the famous Pythagorean theorem, and provides a demonstration of squaring the circle which is quite different from the method of Archimedes who used a spiral or that of Nicomedes who used the conchoid. A Collection manuscript is in the papal library from the 13th century, and is the archetype of all later copies, of which none is earlier than the sixteenth century.

Pesaro, Girolamo Concordia, Commandinus edition stimulated a revival of geometry in the 17th century. The most interesting part of the Collection, measured by its influence on modern mathematics, is Book VII. The collection was reedited by Frederick Hultsch who gave definite Greek text with Latin translation. Thus Descartes demonstrated that the difficulties which Pappus was unable to overcome could be got round by the use of his new algebraic method.

Pappus thus came to play a catalytic, if minor, role in the founding of Cartesian analytical geometry. In his Principia Newton also found inspiration in Pappus; he proved in a purely geometrical manner that the locus with respect to four lines is a conic section, which may degenerate into a circle.

Pappus formally defined analysis and synthesis as they are still commonly applied in the solution of geometrical riders. Pappus stumbled upon the projective invariance of the cross-ratio of four collinear points and other related results reclaimed by modern projective geometry; and he gave the first recorded statement of the focus-directrix property of the three conic sections. He formulated the "centrobaric" theorems, frequently attributed to Paul Guldinfor calculating the volume and surface generated by a plane figure rotating about an axis in its own plane.

He discussed theoretical mechanics, the equilibrium of a heavy body on an inclined plane, the use of the mechanical powers, and the construction of mechanical toys Biographical dictionary of scientists.Other than that he was born at Alexandria in Egypt and that his career coincided with the first three decades of the 4th century adlittle is known about his life.

Judging by the style of his writings, he was primarily a teacher of mathematics. As a source of information concerning the history of Greek mathematics, he has few rivals. His principal work, however, was the Synagoge c. The only Greek copy of the Synagoge to pass through the Middle Ages lost several pages at both the beginning and the end; thus, only Books 3 through 7 and portions of Books 2 and 8 have survived. A complete version of Book 8 does survive, however, in an Arabic translation.

Book 1 is entirely lost, along with information on its contents. The Synagoge seems to have been assembled in a haphazard way from independent shorter writings of Pappus. Nevertheless, such a range of topics is covered that the Synagoge has with some justice been described as a mathematical encyclopedia.

The Synagoge deals with an astonishing range of mathematical topics; its richest parts, however, concern geometry and draw on works from the 3rd century bcthe so-called Golden Age of Greek mathematics. Book 2 addresses a problem in recreational mathematics: given that each letter of the Greek alphabet also serves as a numeral e. Book 4 concerns the properties of several varieties of spirals and other curved lines and demonstrates how they can be used to solve another classical problem, the division of an angle into an arbitrary number of equal parts.

The analytic proof involved demonstrating a relationship between the sought object and the given ones such that one was assured of the existence of a sequence of basic constructions leading from the known to the unknown, rather as in algebra. With three exceptions the books are lost, and hence the information that Pappus gives concerning them is invaluable.

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Pappus of Alexandria

Author of Pappus of Alexandria. See Article History.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. In this respect the fate of Pappus strikingly resembles that of Diophantus.

In his CollectionPappus gives no indication of the date of the authors whose treatises he makes use of, or of the time at which he himself wrote. If we had no other information than can be derived from his work, we should only know that he was later than Claudius Ptolemy whom he often quotes. Suidas states that he was of the same age as Theon of Alexandria, who wrote commentaries on Ptolemy's great work, the Syntaxis mathematicaland flourished in the reign of Theodosius I.

Suidas says also that Pappus wrote a commentary upon the same work of Ptolemy. But it would seem incredible that two contemporaries should have at the same time and in the same style composed commentaries upon one and the same work, and yet neither should have been mentioned by the other, whether as friend or opponent.

It is more probable that Pappus's commentary was written long before Theon's, but was largely assimilated by the latter, and that Suidas, through failure to disconnect the two commentaries, assigned a like date to both. A different date is given by the marginal notes to a 10th-century MS. The question of Pappus's commentary on Ptolemy's work is discussed by Hultsch Pappi collectio Berlin,vol. He 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 MS.

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. Very valuable are the systematic introductions to the various books which set forth clearly in outline the contents and the general scope of the subjects to be treated.

From these introductions we are able to 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. At the same time, his characteristic exactness makes his collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. We proceed to summarize briefly the contents of that portion of the Collection which has survived, mentioning separately certain propositions which seem to be among the most important.

Pappus of Alexandria

We can only conjecture that the lost book i. The whole of book ii. On this subject see Nesselmann, Algebra der Griecken Berlin,pp. Cantor, Gesch. Mathi. Book iii. It may be divided into five sections: i On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates 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.

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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. Of book iv. This and several other propositions on contact, e. Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes already mentioned in book i.Quick Info Born about Alexandria, Egypt Died about Summary Pappus is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.

He wrote commentaries on Euclid's Elements and Ptolemy's Almagest. Biography Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry. Our knowledge of Pappus's life is almost nil. There appear in the literature one or two references to dates for Pappus's life which must be wrong.

There is a reference in the Suda Lexicon a work of a 10 th century Greek lexicographer which states that Pappus was a contemporary of Theon of Alexandria see for example [ 1 ] :- Pappus, of Alexandria, philosopher, lived about the time of the Emperor Theodosius the Elder [ AD - AD ]when Theon the Philosopher, who wrote the Canon of Ptolemyalso flourished. This would seem convincing but there is a chronological table by Theon of Alexandria which, when being copied, has had inserted next to the name of Diocletian who ruled AD - AD "at that time wrote Pappus".

Similar insertions give the dates for PtolemyHipparchus and other mathematical astronomers. Clearly both of these cannot be correct, and the known inaccuracy of the Suda led historians to favour dates for Pappus which would have him writing in the period AD - AD, as suggested by the insertion into Theon 's chronological table.

Pappus of Alexandria

Heath in [ 4 ] is completely convinced saying that [ 4 ] :- Pappus lived at the end of the third century AD. Other than this accurate date we know little else about Pappus. He was born and appears to have lived in Alexandria all his life. We know that he dedicated works to Hermodorus, Pandrosion and Megethion but other than knowing that Hermodorus was Pappus's son, we have no further knowledge of these men. Again Pappus refers to a friend who was also a philosopher, named Hierius, but other than knowing that he encouraged Pappus to study certain mathematical problems, we know nothing else about him either.

Finally a reference to Pappus in Proclus 's writings says that he headed a school in Alexandria. Pappus's major work in geometry is Synagoge or the Mathematical Collection which is a collection of mathematical writings in eight books thought to have been written in around although some historians believe that Pappus had completed the work by AD.

Heath in [ 4 ] describes the Mathematical Collection as follows:- Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field. It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works where still extant rather than to enable them to be dispensed with.

It seems likely that this work was not originally written as a single treatise but rather was written as a series of books dealing with different topics. Each book has its own introduction and often a valuable historical account of the topic, particularly in the case where such an account is not readily available from other sources.

1911 Encyclopædia Britannica/Pappus of Alexandria

Book I covered arithmetic and is lost while Book II is partly lost but the remaining part deals with Apollonius's method for dealing with large numbers. The method expresses numbers as powers of a myriad, that is as powers of Book III is divided by Pappus into four parts.

The first part looks at the problem of finding two mean proportionals between two given straight lines. The second part gives a construction of the arithmetic, geometric and harmonic means.Greek geometer, flourished about the end of the 3rd century A. 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. In this respect the fate of Pappus strikingly resembles that of Diophantus. In his Collection, Pappus gives no indication of the date of the authors whose treatises he makes use of, or of the time at which he himself wrote.

If we had no other information than can be derived from his work, we should only know that he was later than Claudius Ptolemy whom he often quotes. Suidas states that he was of the same age as Theon of Alexandria, who wrote commentaries on Ptolemy's great work, the Syntaxis mathematica, and flourished in the reign of Theodosius I.

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Suidas says also that Pappus wrote a commentary upon the same work of Ptolemy. But it would seem incredible that two contemporaries should have at the same time and in the same style composed commentaries upon one and the same work, and yet neither should have been mentioned by the other, whether as friend or opponent.

It is more probable that Pappus's commentary was written long before Theon's, but was largely assimilated by the latter, and that Suidas, through failure to disconnect the two commentaries, assigned a like date to both. A different date is given by the marginal notes to a loth-century MS. The great work of Pappus, in eight books and entitled 6vvayw'y or Collection, we possess only in an incomplete form, the first book being lost, and the rest having suffered considerably.

The question of Pappus's commentary on Ptolemy's work is discussed by Hultsch,Pappi collectio Berlin,vol. He 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 MS.

Pappus of Alexandria

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. Very valuable are the systematic introductions to the various books which set forth clearly in outline the contents and the general scope of the subjects to be treated. From these introductions we are able to 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.

At the same time, his characteristic exactness makes his collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. We proceed to summarize briefly the contents of that portion of the Collection which has survived, mentioning separately certain propositions which seem to be among the most important.

We can only conjecture that the lost book i. The whole of book ii. On this subject see Nesselmann, Algebra der Griechen Berlin,pp. Cantor, Gesch. Book iii. It may be divided into five sections: 1 On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates 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.

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. Of book iv. At the beginning is the well-known generalization of Eucl.

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This and several other propositions on contact, e. Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes already mentioned in book i.

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 the trisection 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. In book v. 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.Ahmet, Turkey Expedition Voyage around Spitsbergen, June 2012 Hi Cecilia Very professional.

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