Pappus of Alexandria
Encyclopedia
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
s of Antiquity, known for his Synagoge or Collection (c. 340), and for Pappus's Theorem
Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
in projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. Nothing is known of his life, except (from his own writings) that he had a son named Hermodorus, and was a teacher in Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
.
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
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
, doubling the cube
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
, polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s and polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
.
Context
Pappus flourished in the 4th century A.D. 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 DiophantusDiophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
.
Dating
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. If we had no other information, we should only know that he was later than PtolemyPtolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
(died c. 168 AD), whom he quotes, and earlier than Proclus
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
(born c. 411 AD), who quotes him.
The Suda
Suda
The Suda or Souda is a massive 10th century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is an encyclopedic lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost, and often...
(a 10th century Byzantine Greek encyclopedia of known inaccuracy) states that Pappus was of the same age as Theon of Alexandria
Theon of Alexandria
Theon was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's Elements and Ptolemy's Handy Tables, as well as writing various commentaries...
, who flourished in the reign of Emperor Theodosius I
Theodosius I
Theodosius I , also known as Theodosius the Great, was Roman Emperor from 379 to 395. Theodosius was the last emperor to rule over both the eastern and the western halves of the Roman Empire. During his reign, the Goths secured control of Illyricum after the Gothic War, establishing their homeland...
(372–395 AD). 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
Diocletian
Diocletian |latinized]] upon his accession to Diocletian . c. 22 December 244 – 3 December 311), was a Roman Emperor from 284 to 305....
(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
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
he calculates "the place and time of conjunction which gave rise to the eclipse in Tybi
Month of Tobi
Tobi , also known as Touba, is the fifth month of the Coptic calendar. It lies between January 9 and February 7 of the Gregorian calendar...
in 1068 after Nabonassar
Nabonassar
Nabonassar founded a kingdom in Babylon in 747 BC. This is now considered as the start of the Neo-Babylonian Dynasty. At the time the Assyrian Empire was in disarray through civil war and the ascendancy of other kingdoms such as Urartu...
". This works out as October 18, 320 AD, and so we can finally say that Pappus flourished c. 320 AD.
Works
The great work of Pappus, in eight books and entitled Synagoge or Collection, we possess only in an incomplete form, the first book being lost, and the rest having suffered considerably. The SudaSuda
The Suda or Souda is a massive 10th century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Suidas. It is an encyclopedic lexicon, written in Greek, with 30,000 entries, many drawing from ancient sources that have since been lost, and often...
enumerates other works of Pappus. Pappus also wrote commentaries on Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
'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.), and on Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
's Ἁρμονικά (Harmonika).
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. 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, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject.
The whole of Book II (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) related to a system of multiplication due to Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
.
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 ChiosHippocrates of ChiosHippocrates of Chios was an ancient Greek mathematician, , and astronomer, who lived c. 470 – c. 410 BCE.He was born on the isle of Chios, where he originally was a merchant. After some misadventures he went to Athens, possibly for litigation...
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.
- An addition by a later writer on another solution of the first problem of the book.
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.47, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. 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
Arbelos
In geometry, an arbelos is a plane region bounded by a semicircle of diameter 1, connected to semicircles of diameters r and , all oriented the same way and sharing a common baseline. Archimedes is believed to be the first mathematician to study its mathematical properties, as it appears in...
("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
Quadratrix
In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W...
. 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, after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs
Honeycomb conjecture
The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter....
, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus
Zenodorus (mathematician)
- Life and work :Little is known about the life of Zenodorus, although he may have befriended Philonides and made two trips to Athens, as described in Philonides' biography...
'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
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
, 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" (μίκρός άστρονομούμενος), i.e. works other than the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
. It accordingly comments on the Sphaerica of Theodosius
Theodosius of Bithynia
Theodosius of Bithynia was a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere. Born in Tripolis, in Bithynia, Theodosius is cited by Vitruvius as having invented a sundial suitable for any place on Earth...
, the Moving Sphere of Autolycus
Autolycus of Pitane
Autolycus of Pitane was a Greek astronomer, mathematician, and geographer. The lunar crater Autolycus was named in his honour.- Life and work :Autolycus was born in Pitane, a town of Aeolis within Western Anatolia...
, Theodosius's book on Day and Night, the treatise of Aristarchus
Aristarchus of Samos
Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...
On the Size and Distances of the Sun and Moon
Aristarchus On the Sizes and Distances
On the Sizes and Distances is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who flourished circa 280–240 BC...
, and Euclid's Optics and Phaenomena.
Book VII
Since Michel ChaslesMichel Chasles
Michel Floréal Chasles was a French mathematician.He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coalition he was drafted to fight in the defence of Paris in 1814...
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
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, Apollonius
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
, Aristaeus
Aristaeus the Elder
Aristaeus the Elder was a Greek mathematician who worked on conic sections. He was a contemporary of Euclid, though probably older. We know practically nothing of his life except that the mathematician Pappus of Alexandria refers to him as Aristaeus the Elder which presumably means that Pappus was...
and Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...
, 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
Porism
A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem.-Beginnings:...
to theorem and problem. 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
Paul Guldin
Paul Guldin was a Swiss Jesuit mathematician and astronomer. He discovered the Guldinus theorem to determine the surface and the volume of a solid of revolution. This theorem is also known as Pappus–Guldinus theorem and Pappus's centroid theorem, attributed to Pappus of Alexandria...
, 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;
- 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 parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, ellipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, or hyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
according as the constant ratio is equal to, less than or greater than 1 (the first recorded proofs of the properties, which do not appear in Apollonius).
Chasles citation of Pappus was repeated by Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
and Dirk Struik.
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
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
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-ratio
Cross-ratio
In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
s of points and lines. Furthermore, the concept of pole and polar
Pole and polar
In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section...
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 diametersConjugate diameters
In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter...
are given.
Theorems
Although Pappus's Theorem usually refers to Pappus's hexagon theoremPappus's hexagon theorem
In mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
, it may also refer to Pappus's centroid theorem
Pappus's centroid theorem
In mathematics, Pappus' centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution....
.
He also gives his name to the Pappus chain
Pappus chain
In geometry, the Pappus chain was created by Pappus of Alexandria in the 3rd century AD.-Construction:The arbelos is defined by two circles, CU and CV, which are tangent at the point A and where CU is enclosed by CV...
, and to the Pappus configuration and Pappus graph
Pappus graph
In the mathematical field of graph theory, the Pappus graph is a 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon...
arising from his hexagon theorem.
External links
- Pappos (Bibliotheca Augustana)
- "Pappus" in The Columbia Electronic Encyclopedia, Sixth Edition at Answer.com.
- Pappus's Theorem at MathPages
- Pappus's work on the Isoperimetric Problem at Convergence