Pappus quotes from three books of Mechanics and from a work called Barulcus, both by Hero.
Halma) has preserved a fragment, and to which Pappus also refers.
Pappus, in his Collections, treats of its history, and gives two methods by which it can be generated.
Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
He continued his studies in Strassburg, 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 city, and who had great influence over him.
His only extant work is a short treatise (with a commentary by Pappus) On the Magnitudes and Distances of the Sun and Moon.
(iii) Solids of revolution also form a special class, which can be conveniently treated by the two theorems of Pappus (§ 33).
These theorems were discovered by Pappus of Alexandria (c. A.D.
Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus gives lemmas "to the seventh and eighth books" under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth.
The other treatises of Apollonius mentioned by Pappus are - 1st, Aayov alroropii, Cutting off a Ratio; 2nd, Xcopiov a7rorop, Cutting off an Area; 3rd, Ocwpui j Av i Tog, Determinate Section; 4th, 'Eiraci)aL, Tangencies; 5th, 11-€1,o-as, Inclinations; 6th, Tinrot bri ret50t, Plane Loci.
Each of these was divided into two books, and, with the Data, the Porisms and Surface-Loci of Euclid and the Conics of Apollonius were, according to Pappus, included in the body of the ancient analysis.
Pappus gives somewhat full particulars of the propositions, and restorations were attempted by P. Fermat (Ouvres, i., 1891, pp. 3-51), F.
PAPPUS OF ALEXANDRIA, Greek geometer, flourished about the end of the 3rd century A.D.
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.
Suidas says also that Pappus wrote a commentary upon the same work of Ptolemy.
284-305), that Pappus wrote during that period; and in the absence of any other testimony it seems best to accept the date indicated by the scholiast.
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.
Suidas enumerates other works of Pappus as follows: XWpoypacliia obcov i Gepck'iJ, Eis TA 740'6apa Ot13Xia IIToXEµaiov y y6X'Yfs 157r6Avfµa, lroTa/.20US Tob Ev Ats15p, OPECpoKptrtK&.
Pappus himself refers to another commentary of his own on the 'Avfi ojµµa of Diodorus, of whom nothing is known.
These discoveries form, in fact, a text upon which Pappus enlarges discursively.
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.
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.
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.
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.
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.
(4) Der Sammlung des Pappus von Alexandrien siebentes and achtes Buch griechisch and deutsch, published by C. I.
The Geography is a meagre sketch, based mainly on the Chorography of Pappus of Alexandria (in the end of the 4th century), and indirectly on the work of Ptolemy.
Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.
In mathematics he wrote two books On means (IIEpL, Ueuoty) Twp) which are lost, but appear, from a remark of Pappus, to have dealt with " loci with reference to means."
He devised a mechanical construction for two mean proportionals, reproduced by Pappus and Eutocius (Comm.
On the authority of the two great commentators Pappus and Proclus, Euclid wrote four books on conics, but the originals are now lost, and all we have is chiefly to be found in the works of Apollonius of Perga.
Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.
The focus of the parabola was discovered by Pappus, who also introduced the notion of the directrix.
The limb of the calyx may appear as a rim, as in some Umbelliferae; or as pappus, in Compositae and Valeriana.
- Feathery pappus attached to the fruit of Groundsel (Senecio vulgaris).
This pappus is either simple (pilose) or feathery (plumose).