# 1911 Encyclopædia Britannica/Pappus of Alexandria

**PAPPUS OF ALEXANDRIA**, Greek geometer, flourished
about the end of the 3rd century A.D. In a period of general
stagnation in mathematical studies, he stands out as a remarkable
exception. How far he was above his contemporaries,
how little appreciated or understood by them, is shown by the
absence of references to him in other Greek writers, and by the
fact that his work had no effect in arresting the decay of mathematical
science. 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. If we
had no other information than can be derived from his work,
we should only know that he was later than Claudius Ptolemy
whom he often quotes. Suidas states that he was of the same
age as Theon of Alexandria, who wrote commentaries on
Ptolemy's great work, the *Syntaxis mathematical*, and flourished
in the reign of Theodosius I. (A.D. 379-395). Suidas says also
that Pappus wrote a commentary upon the same work of
Ptolemy. But it would seem incredible that two contemporaries
should have at the same time and in the same style
composed commentaries upon one and the same work, and yet
neither should have been mentioned by the other, whether as
friend or opponent. 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.
A different date is given by the marginal notes to a 10th-century
MS., where it is stated, in connexion with the reign of Diocletian
(A.D. 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 συναγωγή
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: Χωρογραφία
οἰκουμενική Εἰς τὰ ....... βιβλία τῆς Πτολεμαίοv neyoXris
......... ........, ........ .... .. ....., bvupoKpiTiKa.
The question of Pappus's commentary on Ptolemy's work is discussed
by Hultsch *Pappi collectio* (Berlin, 1878), vol. iii. p. xiii. seq.
Pappus himself refers to another commentary of his own on the
Ἀνάλημμα of Diodorus, of whom nothing is known. He also
wrote commentaries on Euclid's *Elements* (of which fragments
are preserved in Proclus and the Scholia, while that on the tenth
Book has been found in an Arabic MS.), and on Ptolemy's
'Ἁρμονικά

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. Very valuable are the systematic introductions
to the various books which set forth clearly in outline
the contents and the general scope of the subjects to be treated.
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. At the same time, his characteristic exactness
makes his collection a most admirable substitute for the texts
of the many valuable treatises of earlier mathematicians of
which time has deprived us. We proceed to summarize briefly
the contents of that portion of the *Collection* which has survived,
mentioning separately certain propositions which seem to be
among the most important.

We can only conjecture that the lost book i., as well as book ii., was concerned with arithmetic, book iii. being clearly introduced as beginning a new subject.

The whole of book ii. (the former part of which is lost, the existing
fragment beginning in the middle of the 14th proposition) related
to a system of multiplication due to Apollonius of Perga. On this
subject see Nesselmann, *Algebra der Griecken* (Berlin, 1842), pp.
125-134; and M. Cantor, *Gesch. d. Math*, i.² 331.

Book iii. contains geometrical problems, plane and solid. It may be divided into five sections: (i) On the famous problem of finding two mean proportionals between two given lines, which arose from that of duplicating the cube, reduced by Hippocrates 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. (2) 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. (3) On a curious problem suggested by Eucl. i. 21. (4) On the inscribing of each of the five regular polyhedral in a sphere. (5) An addition by a later writer on another solution of the first problem of the book.

Of book iv. the title and preface have been lost, so that the programme has to be gathered from the book itself. At the beginning
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 ἄρβηλος (*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 B.C., and known by the name
4 TfTpayuvl^ovaa, 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 iukpM iaTpovofmifiems. 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 Porism); (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.

Authorities.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*
*MS _{to} latine versi, accedunt eiusdem de sectione spatii libri duo restitiiti,*

*praemittitur Pappi alexandrini praefatio ad VII*

^{mum}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.