Page:EB1911 - Volume 20.djvu/797

Rh is the well-known generalization of Eucl. 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 (shoemaker’s 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, and known by the name , or quadratrix. 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, 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. Incidentally Pappus describes the thirteen other polyhedral 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.

According to the preface, book vi. is intended to resolve difficulties occurring in the so-called . It accordingly comments on the Sphaerica of Theodosius, the Moving Sphere of Autolycus, Theodosius’s book on Day and Night, the treatise of Aristarchus On the Size and Distances of the Sun and Moon, and Euclid’s Optics and Phaenomena.

The preface of book vii. explains the terms analysis and synthesis, and the distinction between theorem and problem. 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. With the mention of the Porisms of Euclid 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 Guldin, but appear to have been discovered by Pappus himself. Book vii. contains also (I), under the head of the de determinate sectione of Apollonius, lemmas which, closely examined, are seen to be cases of the involution of six points; (2) important lemmas on the Porisms of Euclid (see ); (3) 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 parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than I (the first recorded proofs of the properties, which do not appear in Apllonius).

Lastly, book viii. treats principally of mechanics, the properties of the centre of gravity, and some mechanical powers. Interspersed are some questions of 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 diameters are given.

.Of the whole work of Pappus the best edition is that of Hultsch, bearing the title Pappi alexandrini colleclionis quae supersunt e libris manuscript is edidit latina interpretatione et commentaries inslruxit Fridericus Hultsch (Berlin, 1876-1878). Previously the entire collection had been published only in a Latin translation, Pappi alexandrini mathematical collectiones a Federico Commandino Urbinate in latinum converse et commentariis illustrate (Pesaro, 1588) (reprinted at Venice, 1589, and Pesaro, 1602). A second (inferior) edition of this work was published by Carolus Manolessius.

Of books which contain parts of Pappus’s work, or treat incidentally of it, we may mention the following titles: (1) Pappi alexandrini ''collection es mathematical nunc primum graece edidit Herm. Jos. Eisenmann,''

libri quinti pars altera (Parisiis, 1824). (2) Pappi alexandrini secundi libri mathematical colleclionis fragment um e codice MS. edidit latinum fecit fibtisque illustravil Johannes Wallis (Oxonii, 1688). (3) Apollonii pergaei de sectione rationis libri duo ex arabico MSto latine versi, accedunt eiusdem de sectione spatii libri duo restitiiti, praemittitur Pappi alexandrini praefatio ad VIImum colleclionis mathematical, nunc primum graece edita: cum lemmatibus eiusdem Pappi ad hos Apollonii libros, opera et studio Edmundi Halley (Oxonii. 1706). (4) Der Sammlung des Pappus von Alexandrien siebentes und achtes Buck griechisch und deutsch, published by C. I. Gerhardt, Halle, 1871. (5) The portions relating to Apllonius are reprinted in Heiberg’s Apollonius, ii. 101 sqq.

PAPUANS (Malay papuwah or puwah-puwah, “frizzled,” “woolly-haired,” in reference to their characteristic hairdressing), the name given to the people of New Guinea and the other islands of Melanesia. The pure Papuan seems to be confined to the north-western part of New Guinea, and possibly the interior. But Papuans of mixed blood are found throughout the island (unless the Karons be of Negrito stock), and from Flores in the west to Fiji in the east. The ethnological affinities of the Papuans have not been satisfactorily settled. Physically they are negroid in type, and while tribes allied to the Papuans have been traced through Timor, Flores and the highlands of the Malay Peninsula to the Deccan of India, these “Oriental negroes,” as they have been called, have many curious resemblances with some East African tribes. Besides the appearance of the hair, the raised cicatrices, the belief in omens and sorcery, the practices for testing the courage of youths, &c., they are equally rude, merry and boisterous, but amenable to discipline, and with decided artistic tastes and faculty. Several of the above practices are common to the Australians, who, though generally inferior, have many points of resemblance (osteological and other) with Papuans, to whom the extinct Tasmanians were still more closely allied. It may be that from an indigenous Negrito stock of the Indian archipelago both negroes and Papuans sprang, and that the latter are an original cross between the Negrito and the immigrating Caucasian who passed eastward to found the great Polynesian race.

The typical Papuan is distinctly tall, far exceeding the average Malay height, and is seldom shorter, often taUer, than the European. He is strongly built, somewhat “spur-heeled.” He varies in colour from a sooty-brown to a black, little less intense than that of the darkest negro. He has a small dolichocephalous head, prominent nose somewhat curved and high but depressed at the tip, high narrow forehead with projecting brows, oval face and dark eyes. The jaw projects and the lips are full. His hair is black and frizzly, worn generally in a mop, often of large dimensions, but sometimes worked into plaits with grease or mud. On some islands the men collect their hair into small bunches, and carefully bind each bunch round with fine vegetable fibre from the roots up to within about two inches from the end. Dr Turner gives a good description of this process. He once counted the bunches on a young man’s head, and found nearly seven hundred. There is usually little hair on the face, but chest, legs and fore-arms are generally hirsute, the hair short and crisp.

The constitution of society is everywhere simple. The