# Pappus sentence example

pappus

- 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.Advertisement
- 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.
- They are sometimes known as Guldinus's Theorems, but are more properly described as the Theorems of Pappus.
- 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.Advertisement
- 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.Advertisement
- 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.
- 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.
- 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.Advertisement
- 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.
- 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.
- 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.Advertisement
- 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.Advertisement
- 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.
- Of books which contain parts of Pappus's work, or treat incidentally of it, we may mention the following titles: (I) Pappi alexandrini collectiones mathematicae nunc primum graece edidit Herm.
- (4) Der Sammlung des Pappus von Alexandrien siebentes and achtes Buch griechisch and deutsch, published by C. I.
- In 1749 was published Apollonii Pergaei locorum planorum libri II., a restoration of Apollonius's lost treatise, founded on the lemmas given in the seventh book of Pappus's Mathematical Collection.Advertisement
- 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.Advertisement
- Of the eight books which made up his original treatise, only seven are certainly known, the first four in the original Greek, the next three are found in Arabic translations, and the eighth was restored by Edmund Halley in 1710 from certain introductory lemmas of Pappus.
- 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).Advertisement
- This pappus is either simple (pilose) or feathery (plumose).
- dispersed by the wind on a pappus which develops from the petals.
- The disk seeds have a pappus of hairs for wind dispersal.
- Unlike with creeping thistle, the feathery pappus is attached firmly to the seed.
- Of lost works by Archimedes we can identify the following: (I) investigations on polyhedra mentioned by Pappus; (2) Archai, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) Peri zygon, On balances or levers; (4) Kentrobarika, On centres of gravity; (5) Katoptrika, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) Ephodion, a Method, mentioned by Suidas; (7) Peri sphairopeoia, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens.Advertisement
- (1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm.
- The question of Pappus's commentary on Ptolemy's work is discussed by Hultsch,Pappi collectio (Berlin, 1878), vol.
- In book v., after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus'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.
- -Of the whole work of Pappus the best edition is that of Hultsch, bearing the title Pappi alexandrini collectionis quae supersunt e libris manuscriptis edidit latina interpretatione et commentariis instruxit Fridericus Hultsch (Berlin, 1876-1878).
- In this work (p. 781) it is called "Battata virginiana sive Virginianorum, et Pappus, Potatoes of Virginia."
- In Compositae, Dipsacaceae and Valerianaceae the calyx is attached to the pistil, and its limb is developed in the form of hairs called pappus (fig.
- The third volume includes, however, some theological treatises, and the first part of it is occupied with editions of treatises on harmonics and other works of Greek geometers, some of them first editions from the MSS., and in general with Latin versions and notes (Ptolemy, Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus and Pappus).
- All his numerous other treatises have perished, save one, and we have only their titles handed down, with general indications of their contents, by later writers, especially Pappus.