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Eins ist in der Bibliothek der Akadeniie der Wissenschaften in Berlin, ein zweites war im Besitz von Herrn Professor Heiberg in Kopenhagen, ein drittes, das Suter in Handen hatte, ist nicht mehr nachzuweisen. We designate the binomial, therefore, by addition, (or The line formed by addition) and the apotome by tfuhtro Gtian, (or The line formed by subtraction), because in the former case we add together a medial [area], which is the less, and a rational, which is the greater, whereas in the latter case we subtract the very same medial [area] from the very same ra- tional; and because in the former case we find the line the square upon which is equal to the whole [area] (i.e., the sum of the two areas), whereas in the latter case we find the line the square upon which is equal to the remaining [area] (i. The apotome and the binomial are, therefore, homogeneous, the one being the contrary of the other 77 . A B* is, in this case, greater than BC~ by the square upon a line incommensurable with AB.

Wegen der angegebenen Eigenschaften ist diese Ausgabe natiirlich in den Katalogen schwer zu finden. Now the squares upon AB and BO are equal to twice the rectangle contained by AB and BO plus the square upon AC 74 . Again if the two lines, AB and BC, are commensurable in square, and the sum of the squares upon them is medial, but the rectangle contained by them rational 78 , the medial [area] (i, e., AB 2 |- BC 2 ) is equal to twice the rational (i. Consequently just as we produce the first bimedial by adding a, medial [area] with a rational, granted that the rational is the less and the medial the greater, so, we maintain, the first apotome of a medial is the line the square upon which is equal to the remaining [area] after the subtraction of that rational from that medial.

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m co HARVARD SEMITIC SERIES VOLUMP: vin EDITORIAL, COMMITEE JAMES RICHARD JEWETT, DAVID GORDON LYON, GEORGE FOOT MOORE THE COMMENTARY OF PAPPUS ON BOOK X OF EUCLID'S ELEMENTS LONDON: HUMPHREY MILFORD OXFORD UNIVERSITY PRESS THE COMMENTARY OF PAPPUS ON BOOK X OF EUCLID'S ELEMENTS ARABIC TEXT AND TRANSLATION li Y WILLIAM THOMSON WITH INTRODUCTORY REMARKS, NOTES, AND A GLOSSARY OF TECHNICAL TERMS HY GUSTAV JUNGE AND WILLIAM THOMSON CAMBRIDGE HARVARD UNIVERSITY PRESS 1930 COPYRIGHT, 1930 BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE Druok vou J. (3) Collation of the Arabic Text with the Greek Scholia. WILLIAM THOMSON Bemerkungen zu dem vorliegenden Kommentar. Dieser Kommentar bringt einige Andeutungen liber ein ver- lorenes Werk des Apollonius (I 1, 21, 22, 23; II 1), die WOEPCKE besonders interessierten.

Contents Iajr Vorwort 9 Bernerkungen zu dein vorliegenden Komnientar von Gus TAV J CJNGE 11 Anmerkungen 33 Inhalt des Ivornmentars 35 Introduction by WTXLJAM THOMSON 38 Notes 60 Translation, Part 1 63 Notes 93 Translation, Part II 119 Notes 166 The Arabic Text 189 Notes 261 Glossary 27G LBEMERKUNGEN ZUDEM VORLIEGENDEN KOMMENTAR GUSTAV JITNGK II. (2) The Sources of Pappus's Conception of Rational Quantities. Bald beschaftigte er sich auch mit unserem Kommentar, der die Nummern 5 und 6 des Bandes bildet.

Er mu B sogleich mit dem Studium begonnen haben, jedenfalls erschienen schon 1851 mehrere Abhandlungen von ihm iiber einige dieser Handschrif ten.

In der damaligen Bibliotheque imperiale war ein Sammelbandvon 51 arabischen Handschrif ten, auf den WOEPCKE durch den Katalog aufmerksam wurde. **7 7 wo"Tnust besupphed after *'lneachcafte" (Fi kulli wahidiit ) in the Arabic (W,, p.

Er legte der Acad6mie des Sciences einen Berieht iiber den Kommentar vor, nnter dem Titel: ,, Essai d'une restitution de travaux perdus d' Apollonius sur les quantites irrationelles".

Dieser Berieht enthalt unter anderem ein Verzeichnis der samtlichen Handschriften des Bandes, bringt einige Stiicke des Kommentars, etwa den 15. Teil des gaiizen, arabisch rind in franzosischer Cbersetzung, und den Schlu B der Abhandluiig bildet eine Inhaltsangabe des ganzen Kommentars. 92 I have supplied the words within brackets foj the sake of clarity. WOEPCKE\S conjectures, however, are, from the mathematical point of view, necessary and, from the linguistic pomt of view, quite acceptable (W. Das arabische Manuskript zerfallt in zwei Teile, diese sind aber im Original in keiner Weise weiter gegliedert. 93 The words within brackets, from "The hnc r " to "Jr^.s V'have been sug- gested by WOKPCKE and incorporated in his text, except "Area," which is obviously to be supplied. WOEPCKE'S Essai hat offenbar laiige auf den Druck warten mussen, er erschien erst 185G 1 . Schon 1853 gab CHASLES, der Mathematiker und Historiker der Mathematik, einen Vorbericht iiber WOEPCKE'S Abhandlungen, der kurz und klar sowohl den Gegenstand des Kommentars, namlich das 10. Buch Euklid's, wie auch die Abhandlung WOEPCKE'S charakterisiert 2 . 91 a) Two lines, since as the following sentence informs us, there are two cases of subtraction of a medial from a rational, b) The reason must be sought in the relation of the medial, \/ 6, to the rational, .

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