# The life and works of pappus and his biggest contribution in geometry

Context[ edit ] Pappus was active in the 4th century AD. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. If no other date information were available, all that could be known would be that he was later than Ptolemy died c. This works out as October 18, 320, and so Pappus must have been writing around 320. The Suda enumerates other works of Pappus: The German classicist and mathematical historian Friedrich Hultsch 1833—1908 published a definitive 3-volume presentation of Commandino's translation with both the Greek and Latin versions Berlin, 1875—1878.

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. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. October 2016 Learn how and when to remove this template message 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 whole of Book II the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition discusses a method of multiplication from an unnamed book by Apollonius of Perga. 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 Chios 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. On the inscribing of each of the five regular polyhedra in a sphere. Here Pappus observed that a regular dodecahedron and a regular icosahedron could be inscribed in the same sphere such that their vertices all lay on the same 4 circles of latitude, with 3 of the icosahedron's 12 vertices on each circle, and 5 of the dodecahedron's 20 vertices on each circle.

This observation has been generalised to higher dimensional dual polytopes.

### ADDITIONAL MEDIA

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.

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.

## Pappus of Alexandria

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 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.

- 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;
- Pappus also gives the rather more complicated version of the construction necessary to square the circle;
- As a writer, Pappus must have been quite versatile if the following list of works attributed to him is any indication;
- 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;
- First of all ranks the elegant theorem re-discovered by Guldin , over 1000 years later, that the volume generated by the revolution of a plane curve which lies wholly on one side of the axis, equals the area of the curve multiplied by the circumference described by its centre of gravity;
- Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required when the question involves a greater number of lines.

Book VII[ edit ] Since Michel Chasles cited this book of Pappus in his history of geometric methods, [8] 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 EuclidApolloniusAristaeus and Eratosthenesthirty-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 The life and works of pappus and his biggest contribution in geometry we have an account of the relation of porism 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 Guldinbut 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, including what is called Pappus's hexagon theorem ; 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 conicand is followed by proofs that the conic is a parabolaellipseor hyperbola 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.

Milne [11] gave readers the benefit of his reading of Pappus. 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 Pappus manipulated the complete quadrangleused the relation of projective harmonic conjugatesand displayed an awareness of cross-ratios of points and lines. Furthermore, the concept of pole and polar is revealed as a lemma in Book VII.

Interspersed are some propositions on pure geometry.

- A Short Account of the History of Mathematics 1908 [ edit ] Walter William Rouse Ball Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. Halley , pursued the same track; and thus a number of these ancient treatises were restored, with various success indeed, and with different characters of elegance and accuracy...
- The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus.
- Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third if there be only three ; or to the rectangle of the other two if there be four , or again, that the parallelepiped constructed upon three shall bear a given ratio to that upon the other two and any given line if there be five ; or to the parallelepiped upon the other three if there be six ; or if there be seven that the product obtained by multiplying four of them together shall bear a given ratio to the product of the other three, or if there be eight that the product of four of them shall bear a given ratio to the product of the other four.
- Again, if there be six lines, and if the solid contained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position. He also gives his name to the Pappus chain , and to the Pappus configuration and Pappus graph arising from his hexagon theorem.

Proposition 14 shows how to draw an ellipse through five given points, and Prop. Although Pappus's Theorem usually refers to Pappus's hexagon theoremit may also refer to Pappus's centroid theorem. He also gives his name to the Pappus chainand to the Pappus configuration and Pappus graph arising from his hexagon theorem. Itard 1959 Mathematics And Mathematicians, Vol.

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