I would like to know the dating of this particular manuscript that contains Deuteronomy in the Septuagint version.
How do I find that out? I mean was it written before 3rd century BC or was it written after the NT?
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The way this question is phrased seems to make it a bit complicated to answer.
We can carbon date parchment and ink, with fair accuracy. The exact dating of textual content is far less reliable, and far more complicated than simple physical measurement.
The Codex Sinaiticus is a valuable textual witness for both the Septuagint and the New Testament. The oldest manuscript is known to be classified as "Codex Vaticanus Graecus 1209", containing the books Gen 46:28 – Heb 9:14 is dated as around 350CE. The Codex Sinaiticus is more complete and dated to be of roughly the same age. Sadly, the Old Testament is not nearly complete (Kurt Aland & Barbara Aland: "Der Text des Neuen Testaments. Einführung in die wissenschaftlichen Ausgaben sowie in Theorie und Praxis der modernen Textkritik", Deutsche Bibelgesellschaft: Stuttgart 1981, pp. 117–118.)
The ink for that passage hit the paper in the 4th century but the content of this Septuagint (LXX) Deuteronomy (Dtn) passage (like that content of the New Testament (NT)) was written way before that, by and large at least.
That makes the answer more like: the content was written and transmitted largely unchanged in that form used way before the first parts of the New Testament were written. Since they are now contained within one codex it has to phrased as the content of Dtn predates the NT, but in this book they were written down at the same time.
Deuteronomy is part of the Old Testament, often called the Hebrew Bible as well. But the text in the Codex Sinaiticus (CS) you are inquiring about is not really the same text as the one referred to now as the Hebrew Bible.
The Hebrew Bible is now commonly seen as using the primarily the Masoretic Text (MT) in, well, Hebrew, while the CS uses Dtn from the old Greek translation of the Septuagint (LXX). Both text variants more or less solidified and were canonical – that is not changing that drastically any more – before the first parts of the NT were written down in their earliest variants.
Therefore, if you want to look at the grand scheme, Dtn was written way "before the NT", discounting minor textual variants that might still be found in text that changes in fewer and fewer instances over time. If you want to look at the details, it might be argued that this specific LXX-Dtn used in CS was written at roughly the same time as the rest of this book (cf. interpolations, unique and other textual variants etc.) but not after the NT.
Since most people interested in that text after Jesus were Christian writers and the LXX continued to change a little bit here an there it seems fitting to say that the big core of LXX-Dtn was written down way before, the exact variant seen in CS was written together with the NT of CS at the same time.
Take this with a grain of salt. We do not have a 100% reliable and we do not have 100% complete picture of this either. An exact date is even less likely to emerge any time soon.
Keep in mind that apart from simple errors in copying the source the text were in constant state of lux. This is fundamentally explained with the Geza Vermes' concept of 'rewritten texts' (bad Wikipedia link, better explained by its first use in GV: "Scripture and Tradition in Judaism: Haggadic Studies", In: P. A. H. de Boer (Ed): Studia Post-Biblica. Vol 4. Brill: Leiden, 1973, p. 95 (1. ed: 1961)), more comprehensive and easier accsessible in József Zsengellér: "Rewritten Bible After Fifty Years: Texts, Terms, or Techniques?: A Last Dialogue with Geza Vermes", Supplements to the Journal for the Study of Judaism, Brill: Boston, Leiden, 2014.
Just look at the difficulties the use of LXX quotations by Paul leaves the scholars with:
The problem of identifying the Vorlage of the apostle Paul's frequent quotations from the Hebrew Scriptures has long challenged serious students of Paul's letters. As far back as the 1720's the Englishman William Whiston and the German Johann Carpzov were debating whether the Pauline citations or the Masoretic text more faithfully preserved the wording of the original Hebrew Bible. NT scholarship has long since moved beyond such apologetics to an appreciation of the rich diversity that characterizes Paul's frequent appeals to Scripture. Nevertheless, the fundamental question of the relation between Paul's citations and the known texts of the Hebrew Scriptures has yet to be satisfactorily resolved.
From the standpoint of LXX studies, of course, the more signifiant question is the reverse: how important are Paul's biblical citations as witnesses to the text of the Hebrew Scriptures? Numerous studies have established beyond doubt that Paul drew his quotations from Greek (not Hebrew) biblical texts that stood not far from the mainstream of our present LXX tradition.2 Assuming that the texts of the Pauline citations themselves could be established with reasonable certainty, one might anticipate that they would provide valuable evidence for the text of the Greek Bible in the first century CEo A brief survey of the standard printed editions of the LXX, however, would quickly dispel that notion. The classic edition of Holmes and Parsons appears to exclude the NT citations entirely from its critical apparatus. Drs. Alan Brooke and Norman McLean started out citing the evidence of the NT only when it supported known variants within the LXX manuscript tradition, but shifted to including all "definite quotations" after concluding that the original approach resulted in "a somewhat inadequate treatment of such early and important evidence." The editors of the Gottingen Septuagint have obviously struggled with the same problem, listing Paul's quotations as evidence in certain cases and not in others, with no clear explanation for the variations in treatment.
LXX and NT: A Review; Dietrich Alex Koch. Die Schrift als Zeuge des Evangeliums (ftibingen: J. C. B. Mohr, 1986). In: Bulletin Of The International Organization, For Septuagint And Cognate Studies, Volume 21, Fall, 1988.
There is no "the Septuagint", there never was. Hence it is problematic to assign a concrete date to it.
Although there are many more examples from the extant evidence that I could adduce, the examples I have given are sufficient to demonstrate that the textual tradition of Deuteronomy was not linear (even taking three textual traditions as our starting point) prior to the late first century ce. We can detect groupings of texts around our three major witnesses, but there is enough variation, even among manuscripts that are part of the same textual stream of tradition, that we must acknowledge that variation indicating ongoing scribal work on the text is the rule rather than the exception. Thus we again confront the question, what is the best and most useful way to collate this evidence for scholars, especially those who are not trained as textual critics?
Sidnie White Crawford: "Deuteronomy as a Test Case for an Eclectic Critical Edition of the Hebrew Bible", in Andrés Piquer & Otero Pablo & Torijano Morales: "The Text of the Hebrew Bible and Its Editions. Studies in Celebration of the Fifth Centennial of the Complutensian Polyglot", Brill: Leiden, Boston, 2016, pp. 313–329.
To repeat, inspired from what Nigel J almost wrote in a comment below the question:
The Septuagint attached to the Codex is a variational copy of 'the' Old Testament written in the Septuagint textual tradition, therefore: a Septuagint. As important as it is for a textual witness of the Septuagint tradition, it is an original manuscript, but not 'an original' manuscript of "the" Septuagint, but variation in this lineage. It is therefore written at the same time as the Codex itself and dated the same.