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. 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.
The third part describes a collection of geometrical paradoxes which Pappus says are taken from a work by Erycinus. Other than what is included in this part, we know nothing of Erycinus or his work. The final part shows how each of the five regular polyhedra can be inscribed in a sphere.
The authors of [ 9 ] discuss the muddle Pappus made in Book III of the problem of displaying the arithmetic, geometric and harmonic means of two segments in one circle. Book IV contains properties of curves including the spiral of Archimedes and the quadratrix of Hippias and includes his trisection methods.
Pappus introduces the various types of curves that he will consider:- There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems. Those that can be solved with straight line and circle are properly called 'plane' problems, for the lines by which such problems are solved have their origin in a plane. Those problems that are solved by the use of one or more sections of the cone are called 'solid' problems.
For it is necessary in the construction to use surfaces of solid figures, that is to say, cones. There remain the third type, the so-called 'linear' problem.
For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions. Of this character are the curves discovered in the so-called 'surface loci' and numerous others even more involved These curves have many wonderful properties. More recent writers have indeed considered some of them worthy of more extended treatment, and one of the curves is called 'the paradoxical curve' by Menelaus.
Other curves of the same type are spirals, quadratrices, cochloids, and cissoids. Pappus introduces some of the ideas of Book V by describing how bees construct honeycombs. He concludes his discussion of honeycombs and introduces the aims of his work as follows see for example [ 3 ] or [ 4 ] :- Bees, then, know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each.
But we, claiming a greater share in wisdom than the bees, will investigate a somewhat wider problem, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always the greater, and the greatest of then all is the circle having its perimeter equal to them.
Also in Book V Pappus discusses the thirteen semiregular solids discovered by Archimedes. He compares the areas of figures with equal perimeters and volumes of solids with equal surface areas, proving a result due to Zenodorus that the sphere has greater volume than any regular solid with equal surface area.
He also proves the related result that, for two regular solids with equal surface area, the one with the greater number of faces has the greater volume. As well as reviewing these works, Pappus points out errors which have somehow entered the texts. In Book VII Pappus writes about the Treasury of Analysis see for example [ 3 ] :- The so-called "Treasury of Analysis", my dear Hermodorus, is, in short, a special body of doctrine furnished for the use of those who, after going through the usual elements, wish to obtain power to solve problems set to then involving curves, and for this purpose only is it useful.
Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. Read more on Wikipedia. Since , the English Wikipedia page of Pappus of Alexandria has received more than , page views. His biography is available in 45 different languages on Wikipedia up from 43 in Pappus of Alexandria is the 84th most popular mathematician up from 86th in , the 85th most popular biography from Egypt up from 88th in and the 6th most popular Egyptian Mathematician.
Among mathematicians, Pappus of Alexandria ranks 84 out of In addition to his Synagoge, Pappus is known for Pappus's Theorem in projective geometry.
Virtually nothing is known of his life; even the traditional understanding that he taught at Alexandria cannot be confirmed, and the belief that he had a son named Hermodoros, to whom he dedicated the seventh and eighth books of the Synagoge, is only one possible interpretation of ambiguous language. The earliest surviving copy of Pappus's text, and the basis for all later versions, is Vat. The manuscript seems to have been in the Vatican library by or possibly by , but it does not seem to have been copied until much later.
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