Syntaxis

Pappus

(Πάππος), of Alexandria, the name of one of the later Greek geometers, of whom we know absolutely nothing, beside his works, except the fact that Suidas states him to have lived under Theodosius (A. D. 379-395). From an epigram of the second century, or a little later, in which one Pappus is lauded, Reiske thought that this must be the geometer, who ought, therefore, to be placed in the latter half of the second century. And Harless remarks, in confirmation, that of all the authors named by Pappus, no one is known to have flourished later than the second century. This is but poor evidence, and, on the other hand, the authority of Suidas is by no means of the first order on a point of chronology. We may, therefore, look to other sources of probability, and the only one we can find at all to the purpose is as follows.

Pappus has left a short comment upon a portion of the fifth book of Ptolemy's Syntaxis: or rather of the comment which Suidas states him to have written upon four * This portion is on the fifth book: perhaps the four books were not the first four books. books, nothing is left except a small portion which Theon has preserved and commented on (Syntaxis, Basle, 1538, p. 235 of Theon's Commentary). Now Eutocius mentions Theon and Pappus in the same sentence, as commentators on Ptolemy; and puts them thus together in two different places. This is some presumption against Pappus having been nearly a contemporary of Ptolemy, and in favour of his standing in that relation to Theon. A commentator generally takes an established author, except when the subject of comment is itself a comment, and then he generally takes his own contemporaries. And moreover, those writers who are often named together are more likely than not to be near together in time. The point is of some importance; for Pappus is our chief source of information upon the later history of Greek geometry. It makes much difference as to the opinion we are to form on the decay of that branch of learning, whether the summary which he gives is to be referred to the second or the fourth century. If he lived in the fourth century, it is a very material fact that he could not find one geometer in the two preceding centuries whom he then considered as of note.

Works

Overview

The writings mentioned by Suidas as having come from the pen of Pappus are as follows :-- 1. Μαθηματικῶν συναγωγῶν βιβλία, the celebrated Mathematical Collections, of which we shall presently speak. It is not mentioned by Suidas, but is referred So it is customary to say; but the words of Marinus would admit a suspicion that he refers to a separate commentary on Euclid, written by Pappus. to by Marinus at the end of his preface to Euclid's Data.2. Χορογραφία οἰκουμενική.3. Εἰς τὰ τέσσαρα βιβλία τοῦ Πατολεμαίου μεγάλης Συντάξεως ὑπόμνημα.4. Ποταμοὺς τοὺς ἐν Λιβύῃ.5. Ὀνειροκριτικά. The last four are mentioned by Suidas, and just as here written down in continuous quotation, headed βιβλία δὲ αὐτοῦ.

The Collections

The Collections, as we have them now in print, consist of the last six of eight books. Whether there were ever more than eight is not certain: from the description of his own plan given by Pappus, more might be suspected. No Greek text has been printed: an Oxford The ditty which Savile and Bernard imposed upon that university in the seventeenth century, of printing a large collection of Greek geometry, has been performed hitherto precisely in the order laid down; and the editions of Euclid, Apollonius. and Archimedes, which are the consequence, are confessedly the best products of the press as to their subjects, and in the second case the only one. The next volume was intended to contain Pappus and Theon. edition is long overdue. We cannot make out the negative entirely as to whether the existing Greek manuscripts contain the first and second books: most of them at least do not. Gerard Vossius thought these books lost.

MSS and text

Accounts of the manuscripts will be found in Fabricius (Harless, vol. ix. p. 171), and, with interesting additions, in an appendix to Dr. Wm. Trail's Life of Robert Simson, Bath, 1812, 4to. In the portion which exists the text is as corrupt and mutilated as that of any Greek author who is said to have left more than fragments; and the emendations are sometimes rather inventional than conjectural, if properly named.

Editions

Occasional portions of the Greek text have been published at various times, as follows :--

1. Meibomius, de Proportionibus, Copenhagen, 1655, 4to, p. 155, has given three lemmas from the seventh book (Gr. Lat.).

2. Wallis found in a Savilian manuscript a part of the second book (prop. 16-27), and published it (Gr. Lat.) at the end of his edition of Aristarchus [Oxford, 1688, 8vo.], and again in the third volume of his collected works, Oxford, 1699, folio. The subject of this fragment is the mode of multiplying large numbers; from which it has been suspected that the first two books treated of arithmetic only.

3. Part of the preface of the seventh book is given (Gr. Lat.) by Gregory in the introduction to the Oxford Euclid [EUCLEIDES].

4. The complete preface of the seventh book, with the lemmas given by Pappus, as introductory to the subject of analysis of loci (τοῦ ἀναλυομένου τόπου), are given by Halley (Gr. Lat.), in the preface to his version of Apollonius, de Lectione Rationis, Oxford, 1706, 8vo.

So far Fabricius, verified by ourselves in every case except the part in []: we may add that Dr. Trail gave (op. cit., p. 182) two passages (Gr. Lat.) on the classification of lines, which had been much alluded to by Robert Simson: and that Dr. Trail also states, that in the preface of an edition of Vieta's Apollonius Gallus, 1795, J. G. Camerer gave the Greek of the preface and lemmas relating to Tactions (περὶ ἐπαφῶν). Hoffman and Schweiger mention the second part of the fifth book as published (Gr.) by H. J. Eisenmann, Paris, 1824, folio.

Latin Editions

There are two Latin editions of Pappus. The first, by Commandine, and published by his representatives, was made apparently from one manuscript only. Its description is "Pappi Alexandrini Mathematicae Collectiones a Federico Commandino ....commentariis illustratae," Pisauri, 1588 (folio size, quarto signatures). This edition shows, in various copies, three distinct title pages, the one above, another Venetiis, 1589, a third Pisauri, 1602. It is remarkably erroneous in the paging and the catch-words; but it does happen, we find, that one or the other is correct in every case. There is a cancel which is not found in some copies.

The second edition, by Charles Manolessius, has the same title, augmented, Bononiae, 1660 (larger folio, quarto signatures). It professes to be cleared from innumerable errors. We cannot find any appearance of the use of any additional manuscripts, or any thing except what is usual, namely, correction of obvious misprints and commission of others. And we find that Dr. Trail formed the same judgment. The first edition is the more clearly printed. What Mersenne gives, sometimes called an edition, is a mere synopsis of enlnnciations. An intended edition by John Gallaesius, mentioned by Fabricius, never appeared.

Contents of the Collections

The third book of Pappus treats on the duplication of the cube, geometrical constructions connected with the three kinds of means, the placing in a triangle two lines having a sum together greater than that of the two sides (which was regarded as a sort of wonder), and the inscription of the regular solids in a sphere.

The fourth book treats of various subjects of pure geometry, as also of several extra-geometrical curves, as that called the quadratrix, &c.

The fifth book treats of the properties of plane and solid figures, with reference to the greatest content under given boundaries, &c., at great length.

The sixth book is on the geometry of the sphere.

The seventh book is on geometrical analysis, and is preceded by the curious preface, which, mutilated as it is in parts, is the principal source of information we have on the history and progress of the Greek analysis.

The eighth book is on mechanics, or rather on machines.

Assessment

A great deal might be written on Pappus, with reference to the effect his work has produced on modern geometry by the spirit of inquiry and conjecture which its appearance at once excited. But, unless a full account were given of the contents of the Collections, any such digression would be useless.

Further Information

Suidas; Fabric. Bibl. Gr. vol. ix; Trail, Life of Simson, &c.)

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