The books of the Bibles in early Christianity

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The books of the Bibles in early Christianity"

Transcript

1 Page 1 of 8 The books of the Bibles in early Christianity Author: Jordan Scheetz 1 Affiliation: 1 Biblical and Exegetical Studies Department, Tyndale Theological Seminary, The Netherlands Correspondence to: Jordan Scheetz jordan@tyndale.nl Postal address: Egelantierstraat 1, 1171 JM Badhoevedorp, The Netherlands Dates: Received: 17 Mar Accepted: 14 Sept Published: 07 Mar How to cite this article: Scheetz, J., 2012, The books of the Bibles in early Christianity, HTS Teologiese Studies/Theological Studies 68(1), Art. #1049, 8 pages. hts.v68i A resurgence in the interest in other early Christian literature has brought the issue of the Christian biblical canon(s) to the forefront. Questions in relation to what the literature was, which literature was authoritative, and when did it become authoritative, have all been reopened both on a popular and scholarly level. With this climate, a re-evaluation of primary source information in relation to the various lists was in order. The lists from Origen, Eusebius, the Muratorian Canon, Athanasius, and to a lesser extent Tertullian, were examined. The result was: a nuanced perspective that reflects a three level reading hierarchy that gave precedence to the unquestioned texts, allows for mediated expansion through the questioned texts, and calls for a complete correction of the rejected texts based on the first two levels. Further, although none of the lists are exactly alike, substantial agreement was established between these various lists spanning more than a 150 years. In contrast to Marcion, theological harmony did not appear to be the main consideration in these various lists. Introduction With the publishing of Dan Brown s The Da Vinci Code (2003) and the later released movie, popular culture appears to be fascinated with the possibilities of other Christianities. Although Dan Brown s book is obviously fiction, this curious statement before the prologue lends some sort of credibility to the fictional narrative: All descriptions of artwork, architecture, documents, and secret rituals in this novel are accurate (Brown 2003:1). A statement then from the fictional scholar, Teabing, has some sort of credibility, where he asserts: Constantine commissioned and financed a new Bible, which omitted those gospels that spoke of Christ s human traits and embellished those gospels that made Him godlike. The earlier gospels were outlawed, gathered up, and burned. (Brown 2003:254) With the anachronistic statement in relation to Christ s human traits, this quotation somehow resonates well with modern western culture. Elaine Pagels, Bart Ehrman and Karen King, all American scholars of early Christianity, have written books on a semi-popular level exploring a similar thesis in relation to different texts within early Christianity (Pagels 2004; Ehrman 2005a, 2005b; King 2003). John P. Burgess in reviewing this trend states, These scholars also represent the spirit of 21st-century America, with its love of diversity, its suspicion of traditional authority and its respect for personal experience (Burgess 2004:24). This evaluation may of course be valid in relation to these scholars, but maybe the interest amongst those who buy these books is on a more basic and less subversive level. Could it be that people are just simply not aware that there were other books and differing sizes of authoritative collections? Regardless of the motivations for those who write or buy these books, there seems to be several common assumptions in the modern argument when scholarly and popular positions are conflated. The first assumption is that there were many other books in early Christianity. Secondly, these books at one point were authoritative for Christians before it was decided after several hundred years that only 27 books would be included in the New Testament. Thirdly, these other books represent a broader (less orthodox) view of Jesus. In turn each of these issues will be discussed and evaluated by examining the primary source Christian book lists from circa AD 190 to AD 367, concluding with my own nuanced evaluation of these primary texts and the aforementioned conflated assumptions The Authors. Licensee: AOSIS OpenJournals. This work is licensed under the Creative Commons Attribution License. (Other) books in early Christianity Origen and Eusebius The historical data is quite clear that there were other books in early Christianity. Harry Y. Gamble (1992) states in this regard:

2 Page 2 of 8 Christianity, in turn, produced a large body of its own literature (letters, gospels, narratives of apostolic acts, apocalypses, church orders, etc.), much of which became authoritative for various Christian groups, and so came to be regarded as scripture alongside Jewish scripture. But Christianity did not for a long time attempt to create a canon. (p. 853) In support of this statement there are lists that enumerate certain writings that were considered authoritative in varying degrees. Eusebius (ca ) (Stiewe 2001:927) in 6.25 of his Historia ecclesiastica (Hist. eccl.), quoting Origen ( ,254) (Merlan 2001:2160), states, οὐκ ἀγνοητέον δ εἶναι τὰς ἐνδιαθήκους βίβλους, ὡς Ἑβραῖοι παραδιδόασιν, δύο καὶ εἴκοσι, ὃσος ἀριθμὸς τῶν παρ αὐτοῖς στοιχείων ἐστίν and εἰσὶν δὲ αἱ εἴκοσι δύο βίβλοι καθ Ἑβραίους αἵδε (Eusebius 1932:72). In this list the following English equivalents are given: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, Samuel (1 and 2 as one book), Kings (1 and 2 as one book), Chronicles (1 and 2 as one book), 1 Esdras and Ezra/Nehemiah, Psalms, Proverbs, Ecclesiastes, Song of Songs, Isaiah, Jeremiah (Jeremiah, Lamentations, and the Letter of Jeremiah as one book), Daniel, Ezekiel, Job, and Esther (Eusebius 1932:72). With a further qualifying statement one more book is added, ἔξω δὲ τούτων ἐστὶ τὰ Μακκαβαϊκά (Eusebius 1932:74). Still quoting from Origen, Eusebius goes on, ὡς ἐν παραδόσει μαθὼν περὶ τῶν τεσσάρων εὐαγγελίων, ἃ καὶ μόνα ἀναντιρρητά ἐστιν ἐν τῇ ὑπὸ τὸν οὐρανὸν ἐκκλησίᾳ τοῦ θεοῦ (Eusebius 1932:74). These four are listed as Matthew, Mark, Luke and John. Eusebius prefaces the further quotation from Origen in relation to the letters of the apostles: Καὶ ἐν τῷ πέμπτῳ δὲ τῶν εἰς τὸ κατὰ Ἰωάννην Ἐξηγητικῶν ὁ αὐτὸς ταῦτα περὶ τῶν ἐπιστολῶν τῶν ἀποστόλων φησίν (Eusebius 1932:74). Paul is the first one mentioned but no number is given only that οὐδὲ πάσαις ἔγραψεν αἷς ἐδίδαξεν ἐκκλησίαις, ἀλλὰ καὶ αἷς ἔγραψεν, ὀλίγους στίχους ἐπέστειλεν (Eusebius 1932:74, 76). In relation to Peter it is said, μίαν ἐπιστολὴν ὁμολογουμένην καταλέλοιπεν, ἔστω δὲ καὶ δευτέραν ἀμφιβάλλεται γάρ (Eusebius 1932:76). In relation to John it is said, ἔγραψεν δὲ καὶ τὴν Ἀποκάλυψιν... καταλέλοιπεν καὶ ἐπιστολὴν πάνυ ὀλίγων στίχων, ἔστω δὲ καὶ δευτέραν καὶ τρίτην ἐπεὶ οὐ πάντες φασὶν γνησίους εἶναι ταύτας (Eusebius 1932:76). In a somewhat lengthy discussion of the book of Hebrews it is concluded: οὐ γὰρ εἰκῇ οἱ ἀρχαῖοι ἄνδρες ὡς Παύλου αὐτὴν παραδεδώκασιν. τίς δὲ ὁ γράψας τὴν ἐπιστολήν, τὸ μὲν ἀληθὲς θεὸς οἶδεν, ἡ δὲ εἰς ἡμᾶς φθάσασα ἱστορία ὑπὸ τινῶν μὲν λεγόντων ὅτι Κλήμης, ὁ γενόμενος ἐπίσκοπος Ῥωμαίων, ἔγραψεν τὴν ἐπιστολήν, ὑπὸ τινῶν δὲ ὅτι Λουκᾶς, ὁ γράψας τὸ εὐαγγέλιον καὶ τὰς Πράξεις. (Eusebius 1932:78) If Eusebius s citation of Origen is not overly selective, a particular list of books is found with varying degrees of credibility. Without any doubt are: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, Samuel (1 and 2 as one book), Kings (1 and 2 as one book), Chronicles (1 and 2 as one book), 1 Esdras and Ezra/ Nehemiah, Psalms, Proverbs, Ecclesiastes, Song of Songs, Isaiah, Jeremiah (Jeremiah, Lamentations, and the Letter of Jeremiah, as one book) Daniel, Ezekiel, Job, Esther, Matthew, Mark, Luke, John, an unspecified number of Paul s letters, 1 Peter, 1 John, Revelation, Hebrews (with a question of authorship), and Acts. Books that are listed but with some level of differentiation are: Maccabees, 2 Peter, and 2 and 3 John. Eusebius gives his own discussion in relation to the books of the New Testament in Hist. eccl He prefaces his list with this statement: Εὔλογον δ ἐνταῦθα γενομένους ἀνακεφαλαιώσασθαι τὰς δηλωθείσας τῆς καινῆς διαθήκης γραφάς (Eusebius 1926:256). He lists these books after this initial statement as: Matthew, Mark, Luke, John, Acts, Paul s letters, 1 John, 1 Peter and Revelation (Eusebius 1926:256). These books are summarised with this statement: καὶ ταῦτα μὲν ἐν ὁμολογουμένοις (Eusebius 1926:256). This is even with the previous statement about Revelation where he states, ἐπὶ τούτοις τακτέον, εἴ γε φανείη, τὴν Ἀποκάλυψιν Ἰωαννου (Eusebius 1926:256). The next group of books are prefaced with these words: τῶν δ ἀντιλεγομένων, γνωρίμων δ οὖν ὅμως τοῖς πολλοῖς (Eusebius 1926:256). These books are listed as: James, Jude, and 2 and 3 John. Only 2 and 3 John receive a further description: εἴτε τοῦ εὐαγγελιστοῦ τυγχάνουσαι εἴτε καὶ ἑτέρου ὁμωνύμου ἐκείνῳ (Eusebius 1926:256). He also introduces other books that should be understood in this category: ἐν τοῖς νόθοις κατατετάχθω καὶ τῶν Παύλου Πράξεων ἡ γραφὴ ὅ τε λεγόμενος Ποιμὴν καὶ ἡ Ἀποκάλυψις Πέτρου καὶ πρὸς τούτοις ἡ φερομένη Βαρναβᾶ ἐπιστολὴ καὶ τῶν ἀποστόλων αἱ λεγόμεναι Διδαχαὶ ἔτι τε, ὡς ἔφην, ἡ Ἰωάννου Ἀποκάλυψις, εἰ φανείη ἥν τινες, ὡς ἔφην, ἀθετοῦσιν, ἕτεροι δὲ ἐγκρίνουσιν τοῖς ὁμολογουμένοις. ἤδη δ ἐν τούτοις τινὲς καὶ τὸ καθ Ἑβραίους εὐαγγέλιον κατέλεξαν, ᾧ μάλιστα Ἑβραίων οἱ τὸν Χριστὸν παραδεξάμενοι χαίρουσιν. (Eusebius 1926:256) This expands this second category with these books: the Acts of Paul, the Shepherd of Hermas, the Apocalypse of Peter, Barnabas, the Didache, Revelation, and the Gospel of the Hebrews (Eusebius 1926:256). It is interesting to note why Revelation receives a double listing: ἥν τινες, ὡς ἔφην, ἀθετοῦσιν, ἕτεροι δὲ ἐγκρίνουσιν τοῖς ὁμολογουμένοις (Eusebius 1926:256). The purpose for this list is: διακρίνοντες τάς τε κατὰ τὴν ἐκκλησιαστικὴν παράδοσιν ἀληθεῖς καὶ ἀπλάστους καὶ ἀνωμολογημένας γραφὰς καὶ τὰς ἄλλως παρὰ ταύτας, οὐκ ἐνδιαθήκους μὲν ἀλλὰ καὶ ἀντιλεγομένας, ὅμως δὲ παρὰ πλείστοις τῶν ἐκκλησιαστικῶν γινωσκομένας, ἵν εἰδέναι ἔχοιμεν αὐτάς τε ταύτας καὶ τὰς ὀνόματι τῶν ἀποστόλων πρὸς τῶν αἱρετικῶν προφερομένας. (Eusebius 1926:256, 258) Although it is tempting to understand a radical distinction between the books listed as ὁμολογουμένοις and ἀντιλεγομένοις, the distinction appears rather to be between these first two lists and those writings τῶν αἱρετικῶν. These works are only partially listed and described, ὡς Πέτρου καὶ Θωμᾶ καὶ Ματθία ἢ καί τινων παρὰ τούτους ἄλλων εὐαγγέλια περιεχούσας ἢ ὡς Ἀνδρέου καὶ Ἰωαννου καὶ τῶν ἄλλων ἀποστόλων πράξεις (Eusebius 1926:258). The warrant for the title τῶν αἱρετικῶν is given with a detailed description: ὧν οἰδὲν οὐδαμῶς ἐν συγγράμματι τῶν κατὰ τὰς διαδοχὰς ἐκκλησιαστικῶν τις ἀνὴρ εἰς μνήμην ἀγαγεῖν ἠξίωσεν, πόρρω δέ που καὶ ὁ τῆς φράσεως παρὰ τὸ ἦθος τὸ ἀποστολικὸν ἐναλλάττει χαρακτήρ, ἥ τε γνώμη καὶ ἡ τῶν ἐν αὐτοῖς φερομένων προαίρεσις πλεῖστον ὅσον τῆς ἀληθοῦς ὀρθοδοξίας ἀπᾴδουσα, ὅτι δὴ αἱρετικῶν ἀνδρῶν ἀναπλάσματα τυγχάνει, σαφῶς παρίστησιν. (Eusebius 1926:258) For these reasons this third category of books receives this judgement: ὅθεν οὐδ ἐν νόθοις αὐτὰ κατατακτέον, ἀλλ ὡς ἄτοπα πάντῃ καὶ δυσσεβῆ παραιτητέον (Eusebius 1926:258). Like the list from Origen, Eusebius s own list reflects lists of books with varying degrees of credibility. The first list is

3 Page 3 of 8 those books which are ὁμολογουμένοι : Matthew, Mark, Luke, John, Acts, Paul s letters, 1 John, 1 Peter, and Revelation. The second list is those books which are ἀντιλεγομένοι : James, Jude, 2 and 3 John, the Acts of Paul, the Shepherd of Hermas, the Apocalypse of Peter, Barnabas, the Didache, Revelation, and the Gospel of the Hebrews. The first list appears to represent τὰς γραφάς τῆς καινῆς διαθήκης. The second list, although disputed, merits being listed with the first list. The third list is the books τῶν αἱρετικῶν : the Gospel of Peter, the Gospel of Thomas, the Gospel of Matthias, other similar Gospels of the Apostles, the Acts of Andrew, the Acts of John, and the Acts of other Apostles. These books were not to be counted even as a part of the disputed or spurious books (the second list) evidently because they failed to demonstrate even disputed character; they were viewed as completely unreliable. Eusebius gives another discussion in Hist. eccl In relation to Peter s writings he states: Πέτρου μὲν οὖν ἐπιστολὴ μία, ἡ λεγομένη αὐτοῦ προτέρα, ἀνωμολόγηται, ταύτῃ δὲ καὶ οἱ πάλαι πρεσβύτεροι ὡς ἀναμφιλέκτῳ ἐν τοῖς σφῶν αὐτῶν κατακέχρηνται συγγράμμασιν τὴν δὲ περομένην δευτέραν οὐκ ἐνδιάθηκον μὲν εἶναι παρειλήφαμεν, ὅμως δὲ πολλοῖς χρήσιμος φανεῖσα, μετὰ τῶν ἄλλων ἐσπουδάσθη γραφῶν. τό γε μὴν τῶν ἐπικεκλημένων αὐτοῦ Πράξεων καὶ τὸ κατ αὐτὸν ὠνομασμένον εὐαγγέλιον τό τε λεγόμενον αὐτοῦ Κήρυγμα καὶ τὴν καλουμένην Ἀποκάλυψιν οὐδ ὅλως ἐν καθολικοῖς ἴσμεν παραδεδομένα, ὅτι μήτε ἀρχαίων μήτε μὴν καθ ἡμᾶς τις ἐκκλησιαστικὸς συγγραφεὺς ταῖς ἐξ αὐτῶν συνεχρήσατο μαρτυρίαις. (Eusebius 1926:190, 192) 1 Peter is established as being without any question. 2 Peter is not identified as being undisputed like 1 Peter, but yet has value in being treated with respect, μετὰ τῶν ἄλλων... γραφῶν. These writings are set in contrast with the Acts of Peter, the Gospel of Peter, the Preaching of Peter, and the Revelation of Peter. Eusebius goes on to say, ἀλλὰ τὰ μὲν ὀνομαζόμενα Πέτρου, ὧν μόνην μίαν γνησίαν ἔγνων ἐπιστολὴν καὶ παρὰ τοῖς πάλαι πρεσβυτέροις ὁμολογουμένην, τοσαῦτα (Eusebius 1926:192). Several issues come to the surface when he discusses the writings of Paul: τοῦ δὲ Παύλου πρόδηλοι καὶ σαφεῖς αἱ δεκάτεσσαρες ὅτι γε μήν τινες ἠθετήκασι τὴν πρὸς Ἑβραίους, πρὸς τῆς Ῥωμαίων ἐκκλησίας ὡς μὴ Παύλου οὖσαν αὐτὴν ἀντιλέγεσθαι φήσαντες, οὐ δίκαιον ἀγνοεῖν καὶ τὰ περὶ ταύτης δὲ τοῖς πρὸ ἡμῶν εἰρημένα κατὰ καιρὸν παραθήσομαι. (Eusebius 1926:192) Paul s Epistles are emphatically stated as fourteen in number. Presumably these include: Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, Philemon, and Hebrews. Hebrews is listed with this number with a question in relation to whether or not Paul actually authored the text. Bruce M. Metzger in relation to the absence of Hebrews from Hist. eccl states, Why Eusebius does not mention in his list the Epistle to the Hebrews has been widely discussed; the simplest explanation is that he included it as canonical amongst the Epistles of Paul, which he does not identify one by one (Metzger 1997:205). This passage in Hist. eccl. 3.3 makes this connection explicit. The Acts of Paul are not listed even as a part of the disputed texts. Another text not written by Peter or Paul is mentioned beyond this text with this rational: ἐπεὶ δ ὁ αὐτὸς ἀπόστολος ἐν ταῖς ἐπὶ τέλει προσρήσεσιν τῆς πρὸς Ῥωμαίους μνήμην πεποίηται μετὰ τῶν ἄλλων καὶ Ἑρμᾶ, οὗ φασιν ὑπάρχειν τὸ τοῦ Ποιμένος Βιβλίον, ἰστέον ὡς καὶ τοῦτο πρὸς μέν τινων ἀντιλέλεκται, δι οὓς οὐκ ἄν ἐν ὁμολογουμένοις τεθείη, ὑφ ἑτέρων δὲ ἀναγκαιότατον οἷς μάλιστα δεῖ στοιχειώσεως εἰσαγωγικῆς, κέκριται ὅθεν ἤδη καὶ ἐν ἐκκλησίαις ἴσμεν αὐτὸ δεδημοσιευμένον, καὶ τῶν παλαιτάτων δὲ συγγραφέων κεχρημένους τινὰς αὐτῷ κατείληφα. (Eusebius 1926:192, 194) To summarise this quotation, the Shepherd of Hermas, although rejected by some, is listed because of Hermas s name being mentioned at the end of Romans, its widespread use in the churches, and its quotation by ancient authors. This passage in Hist. eccl. 3.3, like Hist. eccl. 3.25, appears to divide the texts into three different categories. In the first category are those texts in which there is no question in relation to their character: 1 Peter, Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, Philemon, and Hebrews. The second category includes the Shepherd of Hermas which is questionable but still seen as valuable. The third category includes those books that are seen as completely questionable: the Acts of Peter, the Gospel of Peter, the Preaching of Peter, the Revelation of Peter, and the Acts of Paul. Eusebius s different lists make several key issues clear. Eusebius was aware of many books that were present within the larger Christian community. Amongst these books, certain ones were unquestioned in relation to their authenticity and value: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, Samuel (1 and 2 as one book), Kings (1 and 2 as one book), Chronicles (1 and 2 as one book), 1 Esdras and Ezra/Nehemiah, Psalms, Proverbs, Ecclesiastes, Song of Songs, Isaiah, Jeremiah (Jeremiah, Lamentations, and the Letter of Jeremiah, as one book), Daniel, Ezekiel, Job, Esther, Matthew, Mark, Luke, John, Acts, Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, Philemon, Hebrews (with a qualified statement), 1 Peter, 1 John, and Revelation (with a qualified statement). Another group of writings was questioned but still seen as valuable. However, this list appears to be internally inconsistent between the different passages. In Hist. eccl these books are Maccabees, 2 Peter, and 2 and 3 John. In Hist. eccl this second category of books is: James, Jude, 2 and 3 John, the Acts of Paul, the Shepherd of Hermas, the Apocalypse of Peter, Barnabas, the Didache, Revelation (with a qualified statement), and the Gospel of the Hebrews. In Hist. eccl. 3.3 there is only the Shepherd of Hermas. The final group of writings is seen as totally unreliable. In Hist. eccl these books are the Gospel of Peter, the Gospel of Thomas, the Gospel of Matthias, other similar Gospels of the Apostles, the Acts of Andrew, the Acts of John, and the Acts of other Apostles. In Hist. eccl. 3.3 these books are the Acts of Peter, the Gospel of Peter, th e Preaching of Peter, the Revelation of Peter, and the Acts of Paul. What is clear is that

4 Page 4 of 8 there is some cross listing between these last two categories and further that this last category is not exhaustive. Muratorian Canon Albert Sundberg (1973) states succinctly in relation to the Muratorian Canon: As everyone knows, Canon Muratori is a list of New Testament books that was found by Ludovico Antonio Muratori ( ) in the Ambrosian Library at Milan, and is contained in a codex dating from the eighth or possibly the seventh century, which belonged originally to Columban s Monastery at Bobbio. The list of New Testament books is part of this codex, which also contains a collection of tracts and creeds that appeared between the second and fifth centuries and that seem to have been collected and transcribed in the eighth (or seventh) century. (p. 1) Muratori himself dated this statement from circa AD 196 as his title to this fragment indicates: Fragmentum acephalum Caji, ut videtur, Romani Presbyteri, qui circiter Annum Christi 196 floruit, de Canone sacrarum Scripturarum (Muratori 1844:1). Support is derived for this position through the discussion in relation to the Shepherd of Hermas: Pastorem vero nuperrime temporibus nostirs in Urbe Roma Herma conscripsit, sedente Cathedra Urbis Romae Ecclesiae Pio Episcopo fratre ejus (Muratori 1844:2). Sundberg argues that this list comes from the same general time period as Eusebius s and Athanasius s lists as is indicated by the title to his article: Canon Muratori: A Fourth Century List (Sundberg 1973:1). He supports this thesis in large part by his discussion in relation to the Book of Revelation: But both appear to stem from the same milieu of discussion about the canonicity of the Apocalypse of John. And this question of its status finds no sitz im leben in the church until subsequent to Dionysius, and then only in the east. (Sundberg 1973:26) However, there is ample evidence from the dispute with Marcion that at least some, though they were considered heretics, questioned the Book of Revelation at a much earlier time. Eilert Herms (2007) simply presupposes the date and purpose of the fragment as:... am Ende des zweiten Jahrhunderts ein gegen solche Verengungen und Erweiterungen gerichteter gesamtkirchlicher Konsens über den Kreis der im Zentrum der gemeinschaftlichen Weitergabepraxis der Kirche, als im Gottesdienst, zu lesenden und auszulegenden Schriften erreicht ist. (p. 86) In this damaged text, the first book found is described as Tertio Evangelii Librum secundo Lucan (Muratori 1844:1). The second is described as Quarti Evangeliorum Joannis ex discipulis (Muratori 1844:1). Further in relation to John it is said, Quid ergo mirum, si Joannes tam constanter singula etiam in Epistolis suis proferat dicens in semetipso: Quae vidimus oculis nostris, et auribus audivimus, et manus nostrae palpaverunt, haec scripsimus (Muratori 1844:2). Another book is listed as Acta autem omnium Apostolorum sub uno libro scripta sunt Lucas optime Theophile comprehendit (Muratori 1844:2). So far, this list includes Luke, John, 1 John and Acts. With the use of the terms tertio and quarti before the Gospels of Luke and John respectively, one would assume that the Gospels of Matthew and Mark preceded these two texts. Metzger (1997) states: Although the beginning of the list is fragmentary, one can be virtually certain that the Gospel according to Matthew was named first, and that the first line preserved in the Fragment refers to Mark. (p. 195) In a longer quotation Paul s writings are discussed in detail: Epistole autem Pauli, quae, a quo loco, vel qua ex causa directe sint, voluntatibus intelligere, ipse declarant. Primum omnium Corinthiis schisma haeresis inerdicens, deinceps Callactis circumcisionem. Romanis autem ordine Scripturarum, sed et principium earum esse Christum intimans, prolixius scripsit, de quibus singulis necesse est a nobis disputari, cum ipse Beatus Apostolus Paulus sequens praedecessoris sui Johannis ordinem, nonnisi nominatim septem Ecclesiis scribat ordine tali. Ad Corinthios prima, ad Ephesios secunda, ad Philippenses tertia, ad Colossenses quarta, ad Galatas quinta, ad Tessalonicenses sexta, ad Romanos septima. Verum Corinthiis, et Tessalonicensibus licet pro correctione iteretur, una tamen per omnem orbem terrae Ecclesia diffusa esse denoscitur. Et Joannes enim in Apocalypsi licet septem Ecclesiis scribat, tamen omnibus dicit. Verum ad Philemonem una, et ad Titum una, et ad Timotheum duas pro affectu et dilectione, in honore tamen Ecclesiae Catholicae, in ordinatione Ecclesasticae disciplinae sanctificatae sunt. Fertur enim ad Laudecenses, alia ad Alexandrinos Pauli nomine fictae ad haeresem Marcionis; et alia plura, quae in Catholicam Ecclesiam recipi non potest. Fel enim cum melle misceri non congruit. (Muratori 1844:2) This list from Paul includes: Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, and Philemon. The discussion also lists John as the author of Revelation. Two further texts are mentioned as fictae ad haeresem Marcionis, namely letters to the Laodiceans and Alexandrians. The statement close to the end of this passage, et alia plura, quae in Catholicam Ecclesiam recipi non potest, also indicates that there were many other writings viewed similarly to these last two. The discussion continues, distinguishing between different types of writing: Epistola sane Judae, et superscripti Joannis duas in Catholica habentur. Et Sapientia ab amicis Salomonis in honorem ipsius scripta. Apocalypsis etiam Joannis, et Petri, tantum recipimus, quam quidam ex nostris legi in Ecclesia nolunt. (Muratori 1844:2) Jude, 1 and 2 John, and the Wisdom of Solomon are listed as habentur. The Apocalypse of John (Revelation) and the Apocalypse of Peter are listed as recipimus but qualified by the statement quam quidam ex nostris legi in Ecclesia nolunt. The Shepherd of Hermas receives a fair bit of discussion as well: Pastorem vero nuperrime temporibus nostris in Urbe Roma Herma conscripsit, sedente cathedra Urbis Romae Ecclesiae Pio Episcopo fratre ejus. Et ideo legi eum quidem oportet, se publicare vero in Ecclesia Populo, neque inter Prophetas completum numero, neque inter apostolos in finem temporum potest. (Muratori 1844:2)

5 Page 5 of 8 Somehow it receives a verdict on the one hand that et ideo legi eum quidem oportet and on the other se publicare uero in ecclesia populo because it is neque inter Prophetas and neque inter apostolos. It should be read but not published amongst the Prophets or Apostles. The fragment closes with this verdict: Arsinoi autem, seu Valentini, vel Mitiadis nihil in totum recipimus, qui etiam Novum Psalmorum Librum Marcioni concripserunt una cum Basilide Assianum Catafrygum constitutorem (Muratori 1844:2). This increases the list of writings to those of: Arsinoes, Valentinus, Metiades, and a different book of Psalms from Marcion. Like Eusebius, there are different categories for these writings. Those listed with no reservation are Matthew, Mark, Luke, John and Acts. The list from Paul includes Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, Philemon, 1 John, 2 and 3 John, Jude, and the Wisdom of Solomon. Another list gives books that should be read but some think not in the church: the Apocalypse of John (Revelation), the Apocalypse of Peter, and the Shepherd of Hermas. A final group is known of but should not be accepted: letters to the Laodiceans and Alexandrians, and the writings of Arsinoes, Valentinus, Metiades, and a different book of Psalms. Athanasius s 39th Paschal Letter David Brakke (1994) summarises the significance of this document well: In histories of the formation of the Christian biblical canon, the thirty-ninth Festal Letter of Athanasius of Alexandria, written for Easter 367, holds a justifiably prominent place. Not only is this letter the earliest extant Christian document to list precisely the twenty-seven books that eventually formed the generally accepted canon of the New Testament, but Athanasius is also the first Christian author known to have applied the term canonized (κανονιζόμενα) specifically to the books that made up his Old and New Testaments. (pp ) The stated purpose of the letters is in opposition to the heretics: Ἀλλ ἐπειδὴ περὶ μὲν τῶν αἱρετικῶν ἐμνήσθημεν, ὡς νεκρῶν περὶ δὲ ἡμῶν ὡς ἐχόντων πρὸς σωτηρίαν τὰς θείας γραφάς καὶ φοβοῦμαι μήπως, ὡς ἔγραψεν Κορινθίοις Παῦλος, ὀλίγοι τῶν ἀκεραίων ἀπὸ τῆς ἁπλότητος καὶ τῆς ἁγιότητος πλανηθῶσιν, ἀπὸ τῆς πανουργίας τινῶν ἀνθρώπων, καὶ λοιπὸν ἐντυγχάνειν ἑτέρους ἄρξωνται, τοῖς λεγομένοις ἀποκρύφοις, ἀπατώμενοι τῇ ὡμονυμίᾳ τῶν ἀληθῶν βιβλίων. (Athanasius 1844:7) He claims that he is adopting the same attitude as that of τοῦ εὐαγγελιστοῦ Λουκᾶ and is decidedly against those who mix τὰ λεγόμενα ἀπόκρυφα with τῇ θεοπνεύστῳ γραφῇ (Athanasius 1844:7). In relation to the list that follows he claims: ἔδοξεν κᾀμοὶ προτραπέντι παρὰ γνησίων ἀδελφῶν, καὶ μαθόντι, ἄνωθεν ἑξῆς ἐκθέσθαι τὰ κανονιζόμενα καὶ παραδοθέντα πιστευθέντα τε θεῖα εἶναι βιβλία (Athanasius 1844:8). In distinction to this statement from Athanasius, Brakke (1994) argues the following about this list: In any case, Athanasius s polemic against teachers finds its proper context in his effort to reduce the influence of study circles in Christian Alexandria and consolidate Christian life around the hierarchical episcopate. (p. 410) If analogy is to play a role in the assessment of Athanasius s list, this argument, at least in part, finds its weakness in the observation that all of the lists considered thus far span from different time periods, and at least one of them comes from an earlier very influential teacher from Alexandria, namely Origen. Athanasius prefaces the first portion of his list stating, ἔτι τοίνυν τῆς μὲν παλαιᾶς διαθήκης βιβλία τῷ ἀριθμῷ τὰ πάντα εἰκοσιδύο τοσαῦτα γὰρ, ὡς ἤκουσα, καὶ τὰ στοιχεῖα τὰ παρ Ἑβραίοις εἶναι παραδέδονται (Athanasius 1844:8). His list of τῆς παλαιᾶς διαθήκης includes Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, 1 4 Kings (1 2 Samuel and 1 2 Kings), Chronicles (1 2), 1 Esdras and Ezra/Nehemiah, Psalms, Proverbs, Ecclesiastes, Song of Songs, Job, the Twelve, Isaiah, Jeremiah (Jeremiah, Baruch, Lamentations and the Letter of Jeremiah), Ezekiel, and Daniel (Athanasius 1844:8). He prefaces the second portion of the list saying, τὰ δὲ τῆς καινῆς πάλιν οὐκ ὀκνητέον εἰπεῖν (Athanasius 1844:8). His list of τῆς καινῆς includes Matthew, Mark, Luke, John, Acts, James, 1 2 Peter, 1 3 John, Jude, Romans, 1 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 2 Thessalonians, Hebrews, 1 2 Timothy, Titus, Philemon, and Revelation (Athanasius 1844:8 9). He concludes this two part list with a summary statement: ταῦτα πηγαὶ τοῦ σωτηρίου, ὥστε τὸν διψῶντα ἐμφορεῖσθαι τῶν ἐν τούτοις λογίων ἐν τούτοις μόνοις τὸ τῆς εὐσεβαίας διδασκαλεῖον εύαγγελίζεται. μηδεὶς τούτοις ἐπιβαλλέτω μὴ δὲ τούτων ἀφαιρείσθω τι. περὶ δὲ τούτων ὁ κύριος Σαδδουκαίους μὲν ἐδυσώπει, λέγων πλανᾶσθε μὴ εἰδότες τὰς γραφάς. τοῖς δὲ Ιουδαίοις παρῄνει ἐρευνᾶτε τὰς γραφάς ὅτι αὐταί εἰσι αἱ μαρτυροῦσαι περὶ ἐμοῦ. (Athanasius 1844:9) After this strong statement, yet another list of books is introduced: Ἀλλ ἕνεκά γε πλείονος ἀκριβείας προστίθημι καὶ τοῦτο γράφων ἀναγκαίως ὡς ὅτι ἐστὶν καὶ ἕτερα βιβλία τούτων ἔξωθεν οὐ κανονιζόμενα μὲν τετυπωμένα δὲ παρὰ τῶν πατέρων ἀναγινώσκεσθαι τοῖς ἄρτι προσερχομένοις καὶ βουλομένοις κατηχεῖσθαι τὸν τῆς εὐσεβείας λόγον. (Athanasius 1844:9) These books that are οὐ κανονιζόμενα but yet should ἀναγινώσκεσθαι are: the Wisdom of Solomon, the Wisdom of Sirach, Esther, Judith, Tobit, the Didache, and the Shepherd of Hermas (Athanasius 1844:9). The relationship of these three lists is reinforced and contrasted with a fourth category of books: Καὶ ὅμως ἀγαπητοὶ, κᾀκείνων κανονιζομένων καὶ τούτων ἀναγινωσκομένων οὐδαμοῦ τῶν ἀποκρύφων μνήμη (Athanasius 1844:9). The first two lists are books that are κανονιζόμενα, the third list gives those books that should ἀναγινώσκεσθαι, whilst a fourth group is made up of τῶν ἀποκρύφων. This fourth category is described in distinction to these first three lists: ἀλλὰ αἱρετικῶν ἐστιν ἐπίνοια, γραφόντων μὲν ὅτε θέλουσιν αὐτά χαριζομένων δὲ καὶ προστιθέντων αὐτοῖς χρόνους ἵν ὡς παλαιὰ προφέροντες, πρόφασιν ἔχωσιν ἀπατᾷν ἐκ τούτου τοὺς ἀκεραίους (Athanasius 1844:9). Again, the lists of books are separated into different categories. The books that are κανονιζόμενα and are made

6 Page 6 of 8 up of τῆς παλαιᾶς διαθήκης and τῆς καινῆς : Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, 1 4 Kings (1 2 Samuel and 1 2 Kings), Chronicles (1 2), 1 Esdras and Ezra/Nehemiah, Psalms, Proverbs, Ecclesiastes, Song of Songs, Job, the Twelve, Isaiah, Jeremiah (Jeremiah, Baruch, Lamentations, and the Letter of Jeremiah), Ezekiel, Daniel, Matthew, Mark, Luke, John, Acts, James, 1 2 Peter, 1 3 John, Jude, Romans, 1 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 2 Thessalonians, Hebrews, 1 2 Timothy, Titus, Philemon, and Revelation. The books that are οὐ κανονιζόμενα but yet should ἀναγινώσκεσθαι are: the Wisdom of Solomon, the Wisdom of Sirach, Esther, Judith, Tobit, the Didache, and the Shepherd of Hermas. The books that stand outside both of these categories are of τῶν ἀποκρύφων and αἱρετικῶν. Tertullian Tertullian (ca ) (Wlosok 2001:3018), though not listing all the texts, does at least give clues in relation to which texts were viewed as authoritative in relation to the confrontation with Marcion. In Ad Adversus he states: Denique nobis fidem ex apostolis Ioannes et Matthaeus insinuant, ex apostolicis Lucas et Marcus instaurant, isdem regulis exorsi, quantum ad unicum deum attinet creatorem et Christum eius, natum ex virgine, supplementum legis et prophetarum. (Tertullian 1971:262) This is in distinction to Marcion s supposed position in that: cognominatum et ad separationem legis et evangelii coactum, qua duos deos dividens, proinde diversos, alterum alterius intrumenti, vel, quod magis usui est dicere, testamenti, ut exinde evangelio quoque secundum Antitetheses credendo patrocinaretur. (Tertullian 1971:257) This other Gospel is identified by Tertullian in as id evangelium Lucae ab initio editionis suae stare quod cum maxime tuemur, Marcionis vero plerisque nec notum, nullis autem notum ut non eadem damnatum (Tertullian 1971:270). 1 Just before this statement another group of writings was given in : In summa, si constat id verius quod prius, id prius quod et ab initio, id ab initio quod ab apostolis, pariter utique constabit id esse ab apostolis traditum quod apud ecclesias apostolorum fuerit sacrosanctum. Videamus quod lac a Paul Corinthii hauserint, ad quam regulam Galatae sint recorrecti, quid legant Philippenses, Thessalonicenses, Ephesii, quid etiam. Romani de proximo sonent, quibus evangelium et Petrus et Paulus sanguine 1.Dieter T. Roth (2008: ) argues against what he views as the incorrect impression from scholarly debate in Germany in the 1800s over the relation between Marcion s Gospel and Luke. In his opening paragraph he states, Thus, the incorrect impression has arisen that recent advocates of the position that Luke was the product of a significant redactional revision after the time of Marcion are renewing a supposed consensus that resulted from the intense discussion of the issue in Germany 150 years ago (p. 513). Just before his conclusion, he gives this evaluation: Unfortunately, once again, several inaccuracies are present. First, the type of redactional activity seen in Marcion s Gospel and Luke (subsequent to Marcion) if it may even accurately be described as such posited by Ritschl and Volkmar is of a nature vastly different from that set forth by Hilgenfeld, whose view is rather far from Baur s. Secondly, even if one wished to argue that redactional activity of some sort was identified by all four scholars, Ritschl and Volkmar certainly did not conclude that both texts reworked a common original. Finally, not only was there therefore no agreed-upon position or compromise, but it is bewildering that Klinghardt references Ritschl as a proponent of the original text more closely resembling Marcion s Gospel position and Volkmar for the original text more closely resembling Luke position, when both clearly had concluded that Luke, apart from a very few original readings preserved by Marcion, had been edited by Marcion (Roth 2008:526). quoque suo signatum reliquerunt. Habemus et Ioannis alumnas ecclesias. Nam etsi Apocalypsin eius Marcion respuit, ordo tamen episcoporum ad originem recensus in Ioannem stabit auctorem. (Tertullian 1971:268, 270) The books set in opposition to Marcion s books are: the Law, the Prophets, the Gospels (Matthew, Mark, Luke and John), Paul s letters (Romans, Corinthians, Galatians, Ephesians, Philippians and Thessalonians), John s Letters, and Revelation. Tertullian claims that Marcion has a different version of Luke and some of Paul s letters. Authoritative books in early Christianity What is obvious from these different lists is that there was a large body of literature that was circulating within early Christianity. This body of literature contained writings from a variety of recognised time periods, books inherited from the pre-christian era, from the Apostolic-Era, and from the post-apostolic Era. The lists from Origen, Eusebius, the Muratorian Canon and Athanasius distinguish between categories of books rather than just giving one definitive list. These lists move from books that are unquestioned, to those that are questioned, to those that are completely questioned, that is rejected. The books that are seen of value are not simply those in the first category of unquestioned but those in the first two categories, unquestioned and questioned. The books that are rejected are not rejected because there was some question in relation to an aspect of authorship, distribution, or teaching, but instead because they were dubious on all accounts, otherwise they would have been retained with the books that were questioned in relation to one aspect. What is developed is less an exclusive list as a sort of reading hierarchy, where unquestioned books are appropriate for complete use within every aspect of the church, where questioned books are appropriate for personal reading but not as a part of the public gathering of the church, and finally, where completely questioned books are to be rejected as imposters. It has become customary to note that though there are lists, none of these lists are exactly the same. This is of course true, each of these lists are somewhat different from each other. However, the more striking observation in the present milieu is how similar these lists are. The list of books from the Hebrews is only different in regard to one list that excludes Esther (but includes it with the other books to be read) and has the Twelve (instead?). Roger Beckwith (1985) notes in relation to the absence of the Twelve from Origen s list: The omission of the Minor Prophets, whether due to Origen himself or to Eusebius, through whom we receive the list, must be accidental, since their canonicity was never disputed, and Origen both appeals to their authority in his extant writings and wrote a commentary on them, now lost. (p. 186) For the Apostolic literature, all of the Gospels are the same, Paul s Epistles are the same (with a question of authorship with regard to Hebrews), and 1 Peter and 1 John are the same. To use Eusebius s term all of these books are found

7 Page 7 of 8 in the ὁμολογουμένοις. Another striking feature in the present milieu is the second list of books, books that were questioned in some regard but were yet acceptable for some type of use by Christians. Although these lists are not similar, they do contain the books of James, 2 Peter, 2 and 3 John, Revelation, and the Wisdom of Solomon, that were all listed in at least one of the lists of the ὁμολογουμένων. Further, these lists represent a fairly broad age span with the Muratorian Canon coming from the end of the 2nd century AD, Origen s list coming from the mid 3rd century AD, Eusebius s list from the first part of the 4th century AD, and Athanasius s from AD 367. The argument with Marcion and these lists, with multiple levels of reading, also highlight at least some level of broadening in relation to the canon. However, one must note that the lists are different and reflect a different canon than Catholic, Orthodox, and Protestant Bibles. This difference is both hermeneutical as well as substantial. The hermeneutical difference is that some books were considered to be more authoritative than others. This is to say that not all texts have the same status within the life of the church. The unquestioned texts are appropriate for use in every aspect of the church. The questioned texts are for personal reading but not for general use within the church. The rejected books are of no value to the church, whether corporately or personally. This hierarchy of reading gives precedence to the unquestioned texts, allows for mediated expansion through the questioned texts, and calls for a complete correction of the rejected texts based on the first two levels. Substantially this calls for an actual personal examination of texts that in Catholic, Orthodox, and Protestant circles have all but been forgotten. This hierarchy is then not so much between Old and New Testaments and Apocrypha as between these three levels of texts, where the Old and New Testaments as listed in these texts stand on equal unquestioned footing, with the other levels being subject to this category. This is a radically different position than Marcion s as represented by Tertullian where almost all of the unquestioned books from these lists are rejected. This rejection is not based on historical grounds, like these lists, but is hermeneutical. The texts that speak about the God of law are rejected and the ones that speak of the God of gospel are accepted, and at this, one, the Gospel of Luke, is edited from this hermeneutical perspective. This highlights a significant difference between these lists and Marcion s list. Marcion retains books solely based on their harmony in relation to his view of God; the other lists retain books based on their historicity, whether from the Hebrew tradition or apostolic tradition. Robert R. Hann (1977) summarises another harmonistic perspective from the 2nd century AD from the Ebionites found in the socalled Kerygmata Petrou: Among the false passages which the Ebionites believed to have been added to the law are those which portray God in such anthropomorphic terms as experiencing envy, lying, hardening human hearts, or sharing authority with others. Sacrificial worship had not been commanded by God, but was a practice to which the Hebrews were accustomed since their sojourn in Egypt. (p. 236) He states further that: Among the teachings which they reject are the abandonment of the Torah by the Gentile church, its conception that Jesus is divine, its identification of Paul as an apostle, and, perhaps, its doctrine of the trinity. (p. 237) Though the hermeneutic is different from Marcion s and leads to almost the exact opposite collection of books, the principle appears to be the same; books are accepted or rejected based primarily on harmonistic grounds. Martin Ebner in relation to Marcion states, Markions Schriften entsprechen inhaltlich also durchaus der regula fidei (Ebner 2008:47). In the lists discussed, harmony was not the primary consideration but used as a secondary criteria to examine questioned texts; books were rejected only if they failed on historical and then harmonistic accounts. It should be noted further that this harmony was not in relation to other authors in the apostolic tradition but the author to whom the text was connected. This last point is supported by the wide perspectives represented within New Testament literature. Christiane Tietz (2007) makes a similar point whilst arguing for wide diversity in the church: Die Vielgestaltigkeit der Kirche nimmt konkrete Form an in der Mannigfaltigkeit der Konfessionen, die sich alle auf den neutestamentlichen Kanon berufen. Von der Veilstimmigkeit des neutestamentlichen Kanons her ist dazu zu sagen: Keine Konfession hat das Recht zu behaupten, sie allein vertrete die christliche Wahrheit. (p. 102) Conclusion The search through these early lists has confirmed a wide variety of literature that was circulating within early Christianity. Lists that were examined gave judgements in relation to these various books and letters over the period of about 170 years, from circa AD 196 to 367. These lists reveal a varied level of authority, those texts that were unquestioned, those texts that were questioned, and those texts that were rejected. The first level of texts was appropriate for complete use within the church. The second level of texts was appropriate for personal reading but not for public use in the church. The third level of texts was rejected. These lists were based first on historical considerations and second harmonistic considerations were used to evaluate texts that were questioned based on historical considerations. Texts were not rejected because they failed on one account, but rather because they failed on multiple accounts. As was stated in the previous section, This hierarchy of reading gives precedence to the unquestioned texts, allows for mediated expansion through the questioned texts, and calls for a complete correction of the rejected texts based on the first two levels. Substantially this calls for an actual personal examination of texts that in Catholic, Orthodox and Protestant circles have all but been forgotten. This all is in contrast to Marcion s view where harmonistic considerations were of utmost importance. Each of these considerations gives a nuanced evaluation in relation to the present popular and scholarly milieu.

8 Page 8 of 8 Acknowledgements Competing interests The author declares that he has no financial or personal relationship(s) which may have inappropriately influenced him in writing this article. References Athanasius, 1844, Ἐκτη τῆς λθ ἑορταστικῆς ἐπιστολῆς, in J. Kirchhofer (ed.), Quellensammlung zur Geschichte des Neutestamentlichen Canons bis auf Hieronymus, 3rd edn., pp. 7 9, Meyer and Zeller, Zürich. Beckwith, R., 1985, The Old Testament canon of the New Testament church, Wm. B. Eerdmans Publishing Company, Grand Rapids. Brakke, D., 1994, Canon formation and social conflict in fourth-century Egypt: Athanasius of Alexandria s thirty-ninth Festal Letter, Harvard Theological Review 87(4), Brown, D., 2003, The Da Vinci code, Anchor Books, New York. Burgess, J.P., 2004, Alternative Christianities: Going creedless, Christian Century 121(11), Ebner, M., 2008, Der christliche kanon, in M. Ebner & S. Schreiber (Hrsg.), Einleitung in das Neue Testament, pp. 9 52, Kohlhammer, Stuttgart. Ehrman, B.D., 2005a, Lost Christianities: The battles for Scripture and the faiths we never knew, Oxford University Press, Oxford. Ehrman, B.D., 2005b, Lost Scriptures: Books that did not make it into the New Testament, Oxford University Press, New York. Eusebius, 1926, The ecclesiastical history, vol. 1, transl. K. Lake, Harvard University Press, Cambridge, MA. Eusebius, 1932, The ecclesiastical history, vol. 2, transl. J.E.L. Oulton, Harvard University Press, Cambridge, MA. Gamble, H.Y., 1992, Canon: The New Testament, in D.N. Freedman (ed.), The Anchor Bible Dictionary, vol. 1, pp , Doubleday, New York/London/Toronto/ Sydney/Auckland. Hann, R.R., 1977, The undivided way: The early Jewish Christians as a model for ecumenical encounter?, Journal of Ecumenical Studies 14(2), Herms, E., 2007, Die Heilige Schrift als Kanon im kanon tes paradoseos, in B. Janowski (red.), Kanonhermeneutik: Vom Lesen und Verstehen der christlichen Bibel, pp , Neukirchener Verlag, Neukirchen. King, K.L., 2003, The gospel of Mary Magdala: Jesus and the first woman apostle, Polebridge Press, Santa Rosa, CA. Merlan, P., 2001, Origenes, in C. Andresen, H. Erbse, O. Gigon, K. Schefold & K.F. Stoheker (eds.), Lexikon der Antiken Welt, vol. 2, pp , Patmos Verlag, Düsseldorf. Metzger, B.M., 1997, The canon of the New Testament, Oxford Press, New York. Muratori, L.A., 1844, Fragmentum acephalum Caji, ut videtur, Romani Presbyteri, qui circiter Annum Christi 196 floruit, de Canone sacrarum Scripturarum, in J. Kirchhofer (Hrsg.), Quellensammlung zur Geschichte des Neutestamentlichen Canons bis auf Hieronymus, 3rd edn., pp. 1 2, Meyer and Zeller, Zürich. Pagels, E., 2004, Beyond belief: The secret gospel of Thomas, Vintage Books, New York. Roth, D.T., 2008, Marcion s Gospel and Luke: The history of research in current debate, Journal of Biblical Literature 127(3), Stiewe, K., 2001, Eusebios, in C. Andresen, H. Erbse, O. Gigon, K. Schefold & K.F. Stoheker (Hrsg.), Lexikon der Antiken Welt, vol. 1, p. 927, Patmos Verlag, Düsseldorf. Sundberg, A.C. (jr.), 1973, Canon Muratori: A fourth-century list, Harvard Theological Review 66(1), Tertullian, 1971, Adversus Marcionem: Book 4, E. Evans (ed.), Oxford University Press, Oxford, viewed 06 July 2009, from marc/evans_marc_09book4.htm Tietz, C., 2007, Kanon und Kirche, in B. Janowski (Hrsg.), Kanonhermeneutik: Vom Lesen und Verstehen der christlichen Bibel, pp , Neukirchener Verlag, Neukirchen. Wlosok, A., 2001, Tertullian, in C. Andresen, H. Erbse, O. Gigon, K. Schefold & K.F. Stoheker (Hrsg.), Lexikon der Antiken Welt, vol. 3, p. 3018, Patmos Verlag, Düsseldorf.

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

department listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι

department listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι She selects the option. Jenny starts with the al listing. This has employees listed within She drills down through the employee. The inferred ER sttricture relates this to the redcords in the databasee

Διαβάστε περισσότερα

105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support

105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support 105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support These notes were sent to me by Dr. Maurice Robinson and detail 105 places in the Critical Greek Text of the New

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ ΝΟΜΙΚΟ ΚΑΙ ΘΕΣΜΙΚΟ ΦΟΡΟΛΟΓΙΚΟ ΠΛΑΙΣΙΟ ΚΤΗΣΗΣ ΚΑΙ ΕΚΜΕΤΑΛΛΕΥΣΗΣ ΠΛΟΙΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που υποβλήθηκε στο

Διαβάστε περισσότερα

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΨΥΧΟΛΟΓΙΚΕΣ ΕΠΙΠΤΩΣΕΙΣ ΣΕ ΓΥΝΑΙΚΕΣ ΜΕΤΑ ΑΠΟ ΜΑΣΤΕΚΤΟΜΗ ΓΕΩΡΓΙΑ ΤΡΙΣΟΚΚΑ Λευκωσία 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

Section 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016

Section 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Section 1: Listening and Responding/ Aκουστική εξέταση Στο πρώτο μέρος της

Διαβάστε περισσότερα

Modern Greek Extension

Modern Greek Extension Centre Number 2017 HIGHER SCHOOL CERTIFICATE EXAMINATION Student Number Modern Greek Extension Written Examination General Instructions Reading time 10 minutes Working time 1 hour and 50 minutes Write

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

Holy_bible_1 91 923 991 923 993 933 13 961 31 939 3 963 91 92 22 22 22 22 The Greek text here is according to Johannes Kirchhofer, Quellensammlung zur Geschichte des Neutestamentlichen Kanons bis auf

Διαβάστε περισσότερα

Code Breaker. TEACHER s NOTES

Code Breaker. TEACHER s NOTES TEACHER s NOTES Time: 50 minutes Learning Outcomes: To relate the genetic code to the assembly of proteins To summarize factors that lead to different types of mutations To distinguish among positive,

Διαβάστε περισσότερα

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: Discrete-time systems and computer control by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής

Διαβάστε περισσότερα

Instruction Execution Times

Instruction Execution Times 1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11 Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and

Διαβάστε περισσότερα

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ»

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ» I ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΝΟΜΙΚΩΝ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΠΟΛΙΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΟΙΚΟΝΟΜΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ «ΔΙΟΙΚΗΣΗ ΚΑΙ ΟΙΚΟΝΟΜΙΑ» ΚΑΤΕΥΘΥΝΣΗ: ΟΙΚΟΝΟΜΙΚΗ

Διαβάστε περισσότερα

Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS

ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS ΟΜΗΡΟΥ ΙΛΙΑΔΑ ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS www.scooltime.gr [- 2 -] The Project Gutenberg EBook of Iliad, by Homer This ebook is for the use of anyone anywhere at no cost and with almost no restrictions

Διαβάστε περισσότερα

Συστήματα Διαχείρισης Βάσεων Δεδομένων

Συστήματα Διαχείρισης Βάσεων Δεδομένων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

The challenges of non-stable predicates

The challenges of non-stable predicates The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates

Διαβάστε περισσότερα

ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α. Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:.

ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α.  Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:. ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α 2 ειδήσεις από ελληνικές εφημερίδες: 1. Τα Νέα, 13-4-2010, Σε ανθρώπινο λάθος αποδίδουν τη συντριβή του αεροσκάφους, http://www.tanea.gr/default.asp?pid=2&artid=4569526&ct=2 2. Τα Νέα,

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

Démographie spatiale/spatial Demography

Démographie spatiale/spatial Demography ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης

Διαβάστε περισσότερα

1999 MODERN GREEK 2 UNIT Z

1999 MODERN GREEK 2 UNIT Z STUDENT NUMBER CENTRE NUMBER HIGHER SCHOOL CERTIFICATE EXAMINATION 1999 MODERN GREEK 2 UNIT Z (55 Marks) Time allowed Two hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Write your Student

Διαβάστε περισσότερα

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ Μελέτη των υλικών των προετοιμασιών σε υφασμάτινο υπόστρωμα, φορητών έργων τέχνης (17ος-20ος αιώνας). Διερεύνηση της χρήσης της τεχνικής της Ηλεκτρονικής Μικροσκοπίας

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ Ενότητα 1β: Principles of PS Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης

Διαβάστε περισσότερα

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 WEEK 21 The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 I. The Feast of Firstfruits signifies the resurrected Christ as the firstfruits for our enjoyment as

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Finite Field Problems: Solutions

Finite Field Problems: Solutions Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

Galatia SIL Keyboard Information

Galatia SIL Keyboard Information Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing

Διαβάστε περισσότερα

the total number of electrons passing through the lamp.

the total number of electrons passing through the lamp. 1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας»

ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας» ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ ΣΧΟΛΗ ΑΝΘΡΩΠΙΣΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΤΗΣ ΠΡΟΣΧΟΛΙΚΗΣ ΑΓΩΓΗΣ ΚΑΙ ΤΟΥ ΕΚΠΑΙ ΕΥΤΙΚΟΥ ΣΧΕ ΙΑΣΜΟΥ «ΠΑΙ ΙΚΟ ΒΙΒΛΙΟ ΚΑΙ ΠΑΙ ΑΓΩΓΙΚΟ ΥΛΙΚΟ» ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που εκπονήθηκε για τη

Διαβάστε περισσότερα

Right Rear Door. Let's now finish the door hinge saga with the right rear door

Right Rear Door. Let's now finish the door hinge saga with the right rear door Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

Μηχανική Μάθηση Hypothesis Testing

Μηχανική Μάθηση Hypothesis Testing ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ - ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΛ 133: ΑΝΤΙΚΕΙΜΕΝΟΣΤΡΕΦΗΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ ΕΡΓΑΣΤΗΡΙΟ 3 Javadoc Tutorial

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ - ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΛ 133: ΑΝΤΙΚΕΙΜΕΝΟΣΤΡΕΦΗΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ ΕΡΓΑΣΤΗΡΙΟ 3 Javadoc Tutorial ΕΡΓΑΣΤΗΡΙΟ 3 Javadoc Tutorial Introduction Το Javadoc είναι ένα εργαλείο που παράγει αρχεία html (παρόμοιο με τις σελίδες στη διεύθυνση http://docs.oracle.com/javase/8/docs/api/index.html) από τα σχόλια

Διαβάστε περισσότερα

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *2517291414* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2013 1 hour 30 minutes

Διαβάστε περισσότερα

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ EΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΕΚΠΑΙΔΕΥΤΙΚΟ ΤΕΧΝΟΛΟΓΙΚΟ ΙΔΡΥΜΑ ΤΕΙ ΙΟΝΙΩΝ ΝΗΣΩΝ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : Λεωφ. Αντ.Τρίτση, Αργοστόλι Κεφαλληνίας Τ.Κ. 28 100 τηλ. : 26710-27311 fax : 26710-27312

Διαβάστε περισσότερα

Strain gauge and rosettes

Strain gauge and rosettes Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΝΟΜΙΚΗ ΣΧΟΛΗ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΙΣΤΟΡΙΑΣ ΦΙΛΟΣΟΦΙΑΣ ΚΑΙ ΚΟΙΝΩΝΙΟΛΟΓΙΑΣ ΤΟΥ ΔΙΚΑΙΟΥ Διπλωματική εργασία στο μάθημα «ΚΟΙΝΩΝΙΟΛΟΓΙΑ ΤΟΥ ΔΙΚΑΙΟΥ»

Διαβάστε περισσότερα

ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ

ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΕΚΠΑΙΔΕΥΣΗΣ ΣΤΗΝ ΠΡΟΣΧΟΛΙΚΗ ΗΛΙΚΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Διαπολιτισμική Εκπαίδευση και Θρησκευτική Ετερότητα: εθνικές και θρησκευτικές

Διαβάστε περισσότερα

Nuclear Physics 5. Name: Date: 8 (1)

Nuclear Physics 5. Name: Date: 8 (1) Name: Date: Nuclear Physics 5. A sample of radioactive carbon-4 decays into a stable isotope of nitrogen. As the carbon-4 decays, the rate at which the amount of nitrogen is produced A. decreases linearly

Διαβάστε περισσότερα

Business English. Ενότητα # 9: Financial Planning. Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων

Business English. Ενότητα # 9: Financial Planning. Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ Ανώτατο Εκπαιδευτικό Ίδρυμα Πειραιά Τεχνολογικού Τομέα Business English Ενότητα # 9: Financial Planning Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education *4358398658* GREEK 0543/04 Paper 4 Writing May/June 2015 1 hour Candidates answer on the Question

Διαβάστε περισσότερα

*2354431106* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2009

*2354431106* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2009 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *2354431106* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2009 1 hour 30 minutes

Διαβάστε περισσότερα

Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ

Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Η προβολή επιστημονικών θεμάτων από τα ελληνικά ΜΜΕ : Η κάλυψή τους στον ελληνικό ημερήσιο τύπο Σαραλιώτου

Διαβάστε περισσότερα

Advanced Subsidiary Unit 1: Understanding and Written Response

Advanced Subsidiary Unit 1: Understanding and Written Response Write your name here Surname Other names Edexcel GE entre Number andidate Number Greek dvanced Subsidiary Unit 1: Understanding and Written Response Thursday 16 May 2013 Morning Time: 2 hours 45 minutes

Διαβάστε περισσότερα

ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE ΑΠΟ ΑΤΟΜΑ ΜΕ ΤΥΦΛΩΣΗ

ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE ΑΠΟ ΑΤΟΜΑ ΜΕ ΤΥΦΛΩΣΗ ΠΑΝΕΠΙΣΤΗΜΙΟ ΜΑΚΕΔΟΝΙΑΣ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΚΟΙΝΩΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΚΠΑΙΔΕΥΤΙΚΗΣ ΚΑΙ ΚΟΙΝΩΝΙΚΗΣ ΠΟΛΙΤΙΚΗΣ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011 Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι

Διαβάστε περισσότερα

Assalamu `alaikum wr. wb.

Assalamu `alaikum wr. wb. LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump

Διαβάστε περισσότερα

Συντακτικές λειτουργίες

Συντακτικές λειτουργίες 2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.

Διαβάστε περισσότερα

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ Π Τ Υ Χ Ι Α Κ Η Ε Ρ Γ Α Σ Ι Α: Ο ΡΟΛΟΣ ΤΗΣ ΣΥΝΑΙΣΘΗΜΑΤΙΚΗΣ ΝΟΗΜΟΣΥΝΗΣ ΣΤΗΝ ΑΠΟΤΕΛΕΣΜΑΤΙΚΗ ΗΓΕΣΙΑ ΕΠΙΜΕΛΕΙΑ

Διαβάστε περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 1: Elements of Syntactic Structure Το περιεχόμενο του μαθήματος διατίθεται με άδεια

Διαβάστε περισσότερα

SAINT CATHERINE S GREEK SCHOOL ΕΛΛΗΝΙΚΟ ΣΧΟΛΕΙΟ ΑΓΙΑΣ ΑΙΚΑΤΕΡΙΝΗΣ

SAINT CATHERINE S GREEK SCHOOL ΕΛΛΗΝΙΚΟ ΣΧΟΛΕΙΟ ΑΓΙΑΣ ΑΙΚΑΤΕΡΙΝΗΣ SAINT CATHERINE S GREEK SCHOOL ΕΛΛΗΝΙΚΟ ΣΧΟΛΕΙΟ ΑΓΙΑΣ ΑΙΚΑΤΕΡΙΝΗΣ WELCOME TO ST. CATHERINE S GREEK SCHOOL! WEEKDAY GREEK SCHOOL - REGISTRATION FORM SCHOOL YEAR 2016-2017 ΑΙΤΗΣΗ ΕΓΓΡΑΦΗΣ ΣΧΟΛΙΚΗ ΧΡΟΝΙΑ

Διαβάστε περισσότερα

7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple

7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple A/ Ονόματα και ένα παράδειγμα 1 Present Simple 7 Present PERFECT Simple 2 Present Continuous 8 Present PERFECT Continuous 3 Past Simple (+ used to) 9 Past PERFECT Simple she eats she is eating she ate

Διαβάστε περισσότερα

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education GREEK 0543/04 Paper 4 Writing For Examination from 2015 SPECIMEN PAPER Candidates answer on the Question

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

Διαβάστε περισσότερα

Section 9.2 Polar Equations and Graphs

Section 9.2 Polar Equations and Graphs 180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

Section 7.6 Double and Half Angle Formulas

Section 7.6 Double and Half Angle Formulas 09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

Διαβάστε περισσότερα

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education *3148288373* GREEK 0543/04 Paper 4 Writing May/June 2016 1 hour Candidates answer on the Question

Διαβάστε περισσότερα

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014 LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV 18 February 2014 Slowly/quietly Clear/clearly Clean Quickly/quick/fast Hurry (in a hurry) Driver Attention/caution/notice/care Dance Σιγά Καθαρά Καθαρός/η/ο

Διαβάστε περισσότερα

ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα

ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα Ζητήματα υποδοχής και πρόσληψης του Franz Kafka στην Ελλάδα Μιχ. Γ. Μπακογιάννης Aspects of Kafka s reception in Modern Greek Literature: The

Διαβάστε περισσότερα

ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ

ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ Γηπισκαηηθή Δξγαζία ηνπ Φνηηεηή ηνπ ηκήκαηνο Ζιεθηξνιόγσλ Μεραληθώλ θαη Σερλνινγίαο Ζιεθηξνληθώλ

Διαβάστε περισσότερα

F-TF Sum and Difference angle

F-TF Sum and Difference angle F-TF Sum and Difference angle formulas Alignments to Content Standards: F-TF.C.9 Task In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when

Διαβάστε περισσότερα

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ

Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ Νικόλας Χριστοδούλου Λευκωσία, 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education *1880009435* GREEK 0543/04 Paper 4 Writing May/June 2018 1 hour Candidates answer on the Question

Διαβάστε περισσότερα

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Τμήμα Δημοσίων Σχέσεων & Επικοινωνίας Τεχνολογικό Εκπαιδευτικό Ίδρυμα Ιονίων

Διαβάστε περισσότερα

Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author.

Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author. Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author. 2012, Γεράσιμος Χρ. Σιάσος / Gerasimos Siasos, All rights reserved. Στοιχεία επικοινωνίας συγγραφέα / Author

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Ακαδημαϊκός Λόγος Εισαγωγή

Ακαδημαϊκός Λόγος Εισαγωγή - In this essay/paper/thesis I shall examine/investigate/evaluate/analyze Γενική εισαγωγή για μια εργασία/διατριβή Σε αυτήν την εργασία/διατριβή θα αναλύσω/εξετάσω/διερευνήσω/αξιολογήσω... To answer this

Διαβάστε περισσότερα

Notes on the Open Economy

Notes on the Open Economy Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.

Διαβάστε περισσότερα

Αναερόβια Φυσική Κατάσταση

Αναερόβια Φυσική Κατάσταση Αναερόβια Φυσική Κατάσταση Γιάννης Κουτεντάκης, BSc, MA. PhD Αναπληρωτής Καθηγητής ΤΕΦΑΑ, Πανεπιστήµιο Θεσσαλίας Περιεχόµενο Μαθήµατος Ορισµός της αναερόβιας φυσικής κατάστασης Σχέσης µε µηχανισµούς παραγωγής

Διαβάστε περισσότερα

Case 1: Original version of a bill available in only one language.

Case 1: Original version of a bill available in only one language. currentid originalid attributes currentid attribute is used to identify an element and must be unique inside the document. originalid is used to mark the identifier that the structure used to have in the

Διαβάστε περισσότερα

Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς

Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς We review side by side a model of stems ending in ι: πόλις, πόλεως, ἡ = city-state and a masculine model of stems ending in υ: πρέσβυς,

Διαβάστε περισσότερα

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education *8175930111* GREEK 0543/04 Paper 4 Writing May/June 2017 1 hour Candidates answer on the Question

Διαβάστε περισσότερα

HIV HIV HIV HIV AIDS 3 :.1 /-,**1 +332

HIV HIV HIV HIV AIDS 3 :.1 /-,**1 +332 ,**1 The Japanese Society for AIDS Research The Journal of AIDS Research +,, +,, +,, + -. / 0 1 +, -. / 0 1 : :,**- +,**. 1..+ - : +** 22 HIV AIDS HIV HIV AIDS : HIV AIDS HIV :HIV AIDS 3 :.1 /-,**1 HIV

Διαβάστε περισσότερα

Block Ciphers Modes. Ramki Thurimella

Block Ciphers Modes. Ramki Thurimella Block Ciphers Modes Ramki Thurimella Only Encryption I.e. messages could be modified Should not assume that nonsensical messages do no harm Always must be combined with authentication 2 Padding Must be

Διαβάστε περισσότερα

Lecture 2. Soundness and completeness of propositional logic

Lecture 2. Soundness and completeness of propositional logic Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

Διαβάστε περισσότερα

þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼

þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼ Neapolis University HEPHAESTUS Repository School of Health Sciences http://hephaestus.nup.ac.cy Master Degree Thesis 2015 þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼ þÿ Ä Æ Á Â, Á ÃÄ Â þÿ Á̳Á±¼¼±

Διαβάστε περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8  questions or comments to Dan Fetter 1 Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

Νέες ρωγµές στην γερµανική ηγεµονία. (New rifts in the German hegemony) Μαυροζαχαράκης Εµµανουήλ Πολιτικός Επιστήµονας

Νέες ρωγµές στην γερµανική ηγεµονία. (New rifts in the German hegemony) Μαυροζαχαράκης Εµµανουήλ Πολιτικός Επιστήµονας Νέες ρωγµές στην γερµανική ηγεµονία. (New rifts in the German hegemony) Μαυροζαχαράκης Εµµανουήλ Πολιτικός Επιστήµονας Mavrozacharakis Emmanouil Political Scientist Abstract Greek Η προσφυγική κρίση που

Διαβάστε περισσότερα

Test Data Management in Practice

Test Data Management in Practice Problems, Concepts, and the Swisscom Test Data Organizer Do you have issues with your legal and compliance department because test environments contain sensitive data outsourcing partners must not see?

Διαβάστε περισσότερα