V

INTRODUCTORY REMARKS.

beginning of the sixth century, it is beyond doubt that he lived; nor is it likely that he should be placed any later, as he calls Ptolemy "divine" original: "ϑεῖον Ptolemaeum." This was a common honorary title for great authorities in late antiquity..

I did not intend to add those entire Introductory Remarks, or at least the unpublished part, to the works of Diophantus, even though they deserve to be brought to light. Furthermore, the excellent Paris manuscript 2390 codex: a bound book of handwritten pages, which I was able to use, has been corrupted by a more recent hand, and for that reason, there was no opportunity to revisit the Vatican manuscripts. Nevertheless, that fragment might be missed which C. Henry has already published under the title: A small work on sexagesimal multiplication and division to be attributed to Diophantus or Pappus original title: "Opusculum de multiplicatione et divisione sexagesimalibus Diophanto vel Pappo attribuendum." Sexagesimal refers to the base-60 numbering system used in ancient astronomy. (Halle, H. W. Schmidt, 1879). Since the astonishing readings of this editor have more than once frustrated the very sharp Hultsch, not to mention myself ¹), I thought it worthwhile to faithfully describe the same manuscript, namely Paris 453. In this manuscript, Ioannes a Sancta-Maura, around the year 1600, twice inserted that fragment, which he had extracted with some care from a certain Vatican manuscript. I have noted some of Hultsch's conjectures, but anyone who sees the previous edition will immediately understand why I have entirely ignored it.

3. The very learned Heiberg most kindly pointed out to me a third fragment (pages 15 to 31), not recorded in the catalogs, while I was enjoying his company in Paris. The heading "Of Diophantus" original Greek: "Διοφάντου" will deceive no one; here you have writings belonging to Heron original: "Heroniana." Referring to Heron of Alexandria, an engineer and mathematician. in a 14th century manuscript, neither better nor worse than many in Hultsch's collection. At that time, the name of Diophantus had long been almost unknown to the Byzantines, so that

Journal for Mathematics and Physics, Volume 24, Historical-Literary Section, page 199 and following.