What does kαθεξῆς in Luke 1 : 3 mean ? Discovering the writing order of the Gospel of Luke

Copyright: © 2017. The Authors. Licensee: AOSIS. This work is licensed under the Creative Commons Attribution License. Introduction Luke1 states in the preface of his Gospel that he writes it ‘in an orderly account’ (καθεξῆς – 1:3b), but he does not explain what an orderly account means. If καθεξῆς means chronological order, a study of the chronology therein may be important to understand the sequence of events in the Gospel in totality. To study a narrative account, attention may have to be paid to the previous narrative accounts since these accounts may provide additional background information, particularly on the timing of the events. But if καθεξῆς means a logical order, a study of the logic Luke uses may be the key to understand the Gospel. To study a narrative account, we may have to study the reason why Luke places that account in that particular position in the Gospel and what relevance that account has to Luke’s overall logic. Since different beliefs in the writing order may result in different study approaches which may in turn affect the overall understanding of the Gospel and the individual narrative accounts, a study of the writing order Luke uses for his Gospel seems important. This article includes the following sections: Firstly, after summarising the different scholarly views, the article will study all the usages of καθεξῆς in the Bible and ascertain its probable meaning in each occurrence, and will categorise them compared to the categories of meaning shown in BDAG (2000). After that, this article will discuss its etymology to ascertain whether the meaning of its root words impacts καθεξῆς’. Thirdly, this article will study the related words (e.g. synonyms and antonyms) and phrases which share καθεξῆς’ root words to see whether their meanings shed light on the meaning of καθεξῆς. Fourthly, the meanings of καθεξῆς will be discussed in contemporary ancient Greek usages and then be categorised as in section one. The article will end with a conclusion.


Introduction
Luke 1 states in the preface of his Gospel that he writes it 'in an orderly account' (καθεξῆς -1:3b), but he does not explain what an orderly account means.If καθεξῆς means chronological order, a study of the chronology therein may be important to understand the sequence of events in the Gospel in totality.To study a narrative account, attention may have to be paid to the previous narrative accounts since these accounts may provide additional background information, particularly on the timing of the events.But if καθεξῆς means a logical order, a study of the logic Luke uses may be the key to understand the Gospel.To study a narrative account, we may have to study the reason why Luke places that account in that particular position in the Gospel and what relevance that account has to Luke's overall logic.Since different beliefs in the writing order may result in different study approaches which may in turn affect the overall understanding of the Gospel and the individual narrative accounts, a study of the writing order Luke uses for his Gospel seems important.This article includes the following sections: Firstly, after summarising the different scholarly views, the article will study all the usages of καθεξῆς in the Bible and ascertain its probable meaning in each occurrence, and will categorise them compared to the categories of meaning shown in BDAG (2000).After that, this article will discuss its etymology to ascertain whether the meaning of its root words impacts καθεξῆς'.Thirdly, this article will study the related words (e.g.synonyms and antonyms) and phrases which share καθεξῆς' root words to see whether their meanings shed light on the meaning of καθεξῆς.Fourthly, the meanings of καθεξῆς will be discussed in contemporary ancient Greek usages and then be categorised as in section one.The article will end with a conclusion.
2.This scholar supports more than one view.
3.This scholar also supports more than one view.
In the preface of his Gospel Luke says that he is going to write 'in an orderly account' (καθεξῆς).However, scholars have no consensus about the kind of order Luke is seeking.This article aims to ascertain the meaning of καθεξῆς through a detailed study of this word, which includes a study of its usages in the Bible, its etymology, related words, and its meaning in contemporary Greek usages.This article proposes that καθεξῆς most likely refers to chronological order.

The usages of καθεξῆς in the Bible 4
In Brill Dictionary of Ancient Greek (Montanari 2015(Montanari :1002) ) καθεξῆς indicates 'in succession, one after another, in order'.
According to BDAG (2000:490) καθεξῆς has a similar but more elaborative meaning, namely 'pertinent to being in sequence in (1) time, (2) space, or (3) logic'.In the Bible καθεξῆς usually refers to sequence of time, and when Luke uses καθεξῆς to describe time, he describes chronological order, as will be demonstrated.
Kαθεξῆς appears only five times in the New Testament and does not appear in the Septuagint.All five occurrences are by Luke in his Gospel and in Acts.Luke 8:1 describes the beginning of a new episode where Jesus travels with the 12 apostles to towns and villages to preach the gospel, after he has forgiven the sins of a woman in the Pharisee Simon's house (Lk 7:36-50): 'and it happened in the following (days; ἐν τῷ καθεξῆς) 5 also He himself went about through city and village while preaching and bringing good news (about) the kingdom of God and the twelve (were) with Him'.Kαθεξῆς describes a sequence of days in which Jesus' traveling and preaching happens; it also relates the current event in Luke 8:1 to the previous event of 7:36-50 in the sense that the current event happens right after the previous event.It therefore may be categorised as time.
Acts 3:24 is a statement Peter made in his second sermon in Acts: 'and also all the prophets from Samuel and the ones that follow (τῶν καθεξῆς) 6 as many as spoke and proclaimed these days'.Kαθεξῆς here means that follow or following.It describes the prophets who come after Samuel in the order of their appearances in history, which probably refers to chronological order, and therefore fits the category of time.Acts 11:4 describes Peter when he tries to explain to the circumcised believers in Jerusalem the coming of the Holy Spirit to the Gentiles at Cornelius' house: 'But after starting Peter explained to them in chronological order saying' (ἐξετίθετο αὐτοῖς καθεξῆς λέγων). 7Kαθεξῆς refers to in chronological order here because according to the biblical content that follows, Peter describes events in the order in which they happened: he first sees a vision (Ac 11:5-10), then the Holy Spirit tells him to follow the messengers sent by , Cornelius tells him that he is instructed by an angel to send for him (11:13-14), and finally he begins to preach and the Holy Spirit falls on those who listen (11:15).Therefore, this article categorises it as time.

The etymology of καθεξῆς
In this section, a study of the etymology of καθεξῆς will be conducted to understand the meanings of its root words and the impact on the meaning of καθεξῆς when the root words come together.'Etymology' in Collins cobuild advanced learner's English dictionary (CCD; 2003) means 'the study of the origins … of words'.Silva (1994:43-51) believes that the root words of a specific word may not necessarily indicate the real meaning of that word unless the biblical writer intends to use the word according to the meanings of its root words; and such an intention can be demonstrated through the notion of transparency.Silva (ibid:48) further explains that 'the notion of transparency is applied rather broadly to all those words that are motivated, that is, words that have some natural relation to their meaning'.One of the examples Silva gives is the word leader, 'for someone who knows what to lead means and what the suffix -er stands for can easily arrive at the meaning of the word' (ibid:48).Silva (ibid:50) also stresses the ultimate importance of the biblical context: only when the meaning of a word derived from etymology can nicely fit into the biblical context, that the meaning would be meaningful in that specific context.Kαθεξῆς seems to be transparent because on the surface it is a compound consisting of two components (or root words): καθ (the short form of κατά) and ἑξῆς.Since ἑξῆς means sequence in its biblical usages (see my discussion on ἑξῆς below -BWWF), it seems to imply that καθεξῆς is related to some kind of sequence.The meaning of sequence also seems to fit perfectly in the biblical context of Luke 1:3, and it also agrees with the meanings of καθεξῆς in its four other usages in Luke 8:1, Acts 3:24, 11:4 and 18:23, which also indicate sequence.Accordingly, the study of etymology of καθεξῆς seems important.This section shows that καθεξῆς likely means next down to the end, and when Luke writes his Gospel in this manner, he probably writes in chronological order.
There are at least two different opinions about the root words (i.e., the origins) of καθεξῆς.According to Thayer, (1930:313) κατά and ἑξῆς (adverb) are the two root words that form καθεξῆς, while Kohlenberger III et al. (1995:486) opine that κατά and ἔχω (verb) are the two root words of καθεξῆς.Thayer (1930:223) believes that ἑξῆς and ἔχω are related and ἑξῆς comes from the future form of ἔχω (ἕξω).This article will cover all three words: κατά, ἑξῆς and ἔχω.
Kατά is a preposition used extensively in the Greek Bible.(1934:605-606) also agrees that 'the root-meaning of the preposition is not perfectly clear, though "down" seems to be the idea … we sometimes find … the genitive (means) down upon, and the accusative (means) down along.' Robertson also mentions that when κατά is combined with other words, 'often κατά occurs with "perfective" force'.Based on the above observations it is likely that κατά, when combined with ἑξῆς to form καθεξῆς, gives a meaning of down (or indicates a downward motion) and at the same time gives a perfective force to the word, which means down to the end.
Another root of καθεξῆς is the verb ἔχω.The word is frequently used in the Greek Bible and in the NT alone it is used 708 times. 9According to Liddell, Scott & Jones (1996:749-751) ἔχω commonly means to have or to hold.It can also mean 'cling to', 'come next to', 'follow closely', 'neighboring' or 'next '. BDAG (2000:420-422) agrees with LSJ that if ἔχω is used with a spatial aspect it can mean to be next to something' or 'neighboring', and if used with a temporal aspect it can mean 'to be next' or 'immediately following'.These meanings are very similar to the meaning of the adverb ἑξῆς discussed below.
8.Numbers are indicated by the search engine of Bibleworks, v. 7.As the usages are numerous, this article will resort to two reputable lexicons, BDAG and LSJ, for the meaning of the word.9.See Kohlenberger III et al. (1995:n.2398).As the usages are numerous, this article will again resort to BDAG and LSJ for guidance.
Contrary to κατά, the adverb ἑξῆς is not common in the Greek Bible.According to BDAG (2000:349) In the Septuagint ἑξῆς is used six times.In Exodus 10:1 God tells Moses to approach Pharaoh after the seventh plague and God will harden the hearts of Pharaoh and his servants so that he can perform signs amongst the Egyptians.According to the context of 10:1b, ἑξῆς ἐπέλθῃ τὰ σημεῖα represents the remaining three plagues: those that will still come, but not those already been experienced.God has decided to send the remaining plagues on Egypt and this is the reason God asks Moses to approach Pharaoh.Therefore 10:1b may be translated as: 'for I have hardened his heart and (the heart) of his servants so that these successive signs may come upon them'.Eξῆς refers to successive signs which will happen in a time sequence.
In both Deuteronomy 2:34 and 3:6 ἑξῆς is used to describe how the Israelites destroy the cities they have conquered in Canaan (ἐξωλεθρεύσαμεν πᾶσαν πόλιν ἑξῆς).This article agrees with Brenton (1998Brenton ( -1999) that ἑξῆς should be translated as in succession.Eξῆς seems to indicate a spatial sequence for the cities which are destroyed by the Israelites.Judges 20:48 describes how the Israelites destroy the cities and the people of the Benjaminites.Judges 20:48b may be translated literally as: 'and they smote them with edge of sword from (the) next city to the (last) animal, to all things which were found …' (ἀπὸ πόλεως ἑξῆς ἕως κτήνους ἕως).Again, ἑξῆς is used to describe the spatial sequence of the cities of the Benjaminites which are destroyed one by one.
Eξῆς is used in 2 Maccabees 7:8, which describes the seven brothers killed one by one by King Antiochus as he tries to force them to give up their Jewish beliefs.It may be translated as 'and he, having answered in the language of his father, replied: No! For this very reason also he received the next torture as the first (brother)' (οὗτος τὴν ἑξῆς ἔλαβεν βάσανον ὡς ὁ πρῶτος).Eξῆς describes the death of the brothers in a time sequence.In 3 Maccabees 1:9 ἑξῆς describes Ptolemy entering Jerusalem: 'and after (Ptolemy) arriving in Jerusalem and after sacrificing to the great God and after paying gifts and after doing something of the next acts in the place' (καὶ τῶν ἑξῆς τι τῷ τόπῳ ποιήσας).Eξῆς describes a sequence of acts done by Ptolemy in chronological order according to their levels of importance in the Jewish custom.Sacrificing to the great God seems to be the most important act; paying gifts apparently is the next most important one, and the next acts likely are the least important ones, so the author does not even describe them.Therefore this article categorises it as a chronological sequence of events which is related to hierarchy of significance.
From the above, the meanings of ἑξῆς in the Septuagint can be divided into three categories.

The words related to καθεξῆς
In this section the article will explore the meanings of the words related to καθεξῆς and also the phrases in the NT and the Septuagint which contain the root words of καθεξῆς.To understand whether a word conveys a concept (and in this case whether καθεξῆς conveys the concept of sequence or chronological order), Long (2005:146) suggests that it is problematic only to study the word: 'in reality, a single theme may be represented by several different words or phrases, and the totality of a certain truth/theme may not be represented in one word', and he proposes further that one also has to study the word's cognates (words built from the same Greek root), synonyms (other words having similar meanings), and even antonyms (words with the opposite meaning).Therefore, comparing the meaning of these words and phrases may yield a more accurate understanding of the meaning of καθεξῆς in Luke 1:3.For the first part of this exercise, related or cognate words are defined as follows: firstly, words which have the same roots as καθεξῆς, i.e. words formed by κατά and ἑξῆς or κατά and ἔχω; secondly words in the same word family as classified by GECNT, including antonyms (words opposite in meaning to καθεξῆς) and synonyms (words with meaning similar to καθεξῆς); and thirdly, words indicated by BDAG or LSJ as having a meaning similar to and used interchangeably with καθεξῆς.For the second part, related phrases are defined as follows: A related phrase should include either κατά or its related form καθ and then ἑξῆς in proper sequence, or secondly κατά or its related form καθ and then ἔχω or its related forms in proper sequence.
Using these guidelines, cognate words are κατέχω, κάθεξις, ἀνοχή and ἐφεξῆς.However, a detailed analysis of κατέχω,

The meaning of καθεξῆς in contemporary Greek usage
In this section the article studies καθεξῆς in contemporary Greek usage to ascertain its possible meaning(s) during the time of Luke: firstly all uses of καθεξῆς cited by BDAG (2000) and LSJ, and in the Apocrypha, the Pseudepigrapha, Josephus and Philo; secondly all uses in the papyri; and thirdly all additional uses from the last century bce to ce first century in the Thesaurus Linguae Graecae (TLG) electronic data bank. 15This article categorises the meaning of each usage according to the three categories mentioned earlier, i.e. either a sequence related to time, distance or space, or logic or hierarchy of significance. 16This article discovered that καθεξῆς does not refer to any logical order, but to a sequence which means 'next down to the end' referring to a sequence of time, or space, or a hierarchy of significance.
11.In the study of words related to καθεξῆς, κατέχω is identified as having the same root words (κατά and ἔχω).An analysis of the meanings of all the 68 appearances (from GECNT and Hatch & Redpath 1998) of κατέχω and its related forms in the NT and the Septuagint finds that the meaning of the word is not relevant to καθεξῆς.This word seems only to pick up the meaning of to have and to hold from ἔχω (instead of the meaning next) and means hold, hold back, hold fast, suppress or bound, et cetera.There is another word, κάθεξις, which according to LSJ (1996:852), is derived from κατέχω and probably shares the same roots.This word does not appear in either the NT or the Septuagint.According to LSJ it has a meaning similar to κατέχω which is also irrelevant to the meaning of καθεξῆς.
12.In the study of words that belong to the same word family as καθεξῆς, GECNT does not show any related words for καθεξῆς.It only indicates that κατά and ἔχω are the two root words for καθεξῆς (and as mentioned above, Thayer opines that κατά and ἑξῆς are the two root words).Therefore, this exercise focused on the root words ἔχω and ἑξῆς, which provide the meaning next to καθεξῆς.While GECNT indicates that the only related word for ἑξῆς is ἔχω, there are 48 related words for ἔχω.A detailed study of all these words (which include καθεξῆς, κατέχω and ἑξῆς) shows that the only seemingly relevant word is ἀνοχή, which is formed by the root words ἀνα (meaning: up; the opposite of κατά meaning: down) and ἔχω.However, a study of ἀνοχή shows that it picks up the meaning of to hold and to have rather than the meaning of next from ἔχω, and means holding back or forbearance, which is irrelevant to the meaning of καθεξῆς.
13.Using the search engine of Bibleworks v. 7, searches for the following related phrases were conducted for both the NT and the Septuagint.(1) κατά (or καθ) εχ* phrased together in proper sequence; and (2) κατά (or καθ) εξ* phrased together in proper sequence; and (3) κατά (or καθ) εσ* phrased together in proper sequence; (4) κατά (or καθ) ἑξῆς phrased together in proper sequence.Nothing relevant to the meaning of καθεξῆς is found.Another search was conducted for any of the above four combinations not phrased together, but appearing as close neighbours in the same verse.Again, nothing relevant to the meaning of καθεξῆς was found.
15.No usage of καθεξῆς is found in several reputable additional references.These references include: Deissmann (1995); Grenfell andHunt (1898-1994;all 18 volumes, except volume 4 which cannot be located, were checked); Horsely (1981Horsely ( -1989)); and Moulton andMilligan (1930: §2517) ; 16.The categorisations of the meanings of καθεξῆς in this section and in the study of καθεξῆς in the Bible are summarised in Appendix 1 for easy reference.

References cited in BDAG, LSJ and Pseudepigrapha
No usage of καθεξῆς was found in Josephus, Philo or the Apocrypha, but eight usages of καθεξῆς appear in BDAG, LSJ and the Pseudepigrapha.This describes a group singing and the use of a myrtle branch in the group: but others say the myrtle branch is not to proceed in the order from one to one's neighbor in the same couch, but is to pass across according to each (position) from couch to couch.
Paul A. Clement (ed.Lake 1912Lake -1913) ) translates καθεξῆς as 'from each guest to his neighbor in orderly sequence'.The way the myrtle branch proceeds is described later in more detail by Plutarch in the same paragraph.Clement translates it as follows: that the first man to sing sent it over to the first man on the second couch, and the latter to the first man on the third couch, then the second man to the second on the neighboring couch, and so on; so, they say, it seems that the song was named scolium because of the intricate and twisted character of its path.
Plutarch does not treat the way the branch is actually passed along as καθεξῆς, though it is also a logical order.Kαθεξῆς, in Plutarch's understanding, seems to mean a spatial sequence which features the characteristic 'next down to the end' (meaning of καθεξῆς found in its etymology); while next refers to the neighbour closest to its predecessor in the unit of measure explicitly mentioned or implied, and shares the same subject under discussion 19 with its predecessor.(In a time sequence, unit of measure refers to time; in a spatial sequence, unit of measure refers to distance; in a sequence of hierarchy of importance, unit of measure refers to level of importance.)In this case, next refers to the next person sitting on the same couch -the one who is the closest in distance (unit of measure) to the predecessor and they share the same couch (subject under discussion).The branch is passed from the first person on a couch to the second (next) person on the same couch and then to the third (again: next) and all the way (down) until it reaches the last person on the couch (to the end).Therefore, καθεξῆς does not refer to logical order; it refers to a spatial sequence which has the characteristic 'next down to the end'.This describes a large wedding conducted during the time of Alexander the Great.N.G.Wilson has rightly translated the passage as follows: 'When Alexander captured Darius he celebrated his own marriage and that of his friends.The number of people marrying was ninety … For five days in succession he celebrated the weddings (πέντε δὲ ἡμέρας καθεξῆς τοὺς γάμους ἔθυεν) ...' Aelian used καθεξῆς to describe how these five days are organised: it is five days in a row from the first day of the wedding to the second (next) day and all the way (down) to the fifth day (to the end).Kαθεξῆς here also features the characteristic of 'next down to the end', with next referring to the next day, the one closest in time (the unit of measure) to the previous day in the same wedding (subject under discussion).Kαθεξῆς refers to a sequence which describes time.

Inscriptiones Graecae ad res Romanas Pertinentes ('Greek inscriptions relating to Roman affairs' IGR IV [1927,1432: §9])
The phrase describes the names of contest winners engraved in an inscription.The Greek may be translated as 'the contests after having written under an inscription: Smyrna, Olympia in the sixth and tenth, Hadrian Rome second, Puteoli second, Neapolis third, Actia second, the body of men-at arms from Argos, Nemea third, all (winners) in their winning order' (πάντας καθεξῆς).Kαθεξῆς here refers to the order of winning contestants (viz.champion, first runner up, etc.).Kαθεξῆς describes a hierarchy of significance of the success of the performance of the cities in the contests, beginning with the cities which won first prize, then the cities which won second prize (next), and so on (down) until all the winners are described (to the end).Next here refers to the next award of importance (unit of measure) in the same contest (subject under discussion).

Oppianus Apamensis Epicus' Cynegetica 3.59 20
It describes the several pregnancies of a lioness.A.W. Mair (Oppianus 1928) translates the passage as follows: Five times doth the lioness loose her zone in birth, and idle truly is the report that she bears but one.Five she bears the first time, but next she travails with four cubs; then next in order (κατά θ' ἑξείης) from her third labor spring three; from her fourth spring twin young; and last from her womb of noble progeny the mother brings forth the glorious Lion King.
Kατά θ' ἑξείης, which includes the two root words κατά and ἑξείης, may be translated as down next or next down. 21Similar to καθεξῆς, this phrase describes a sequence meaning 'next down to the end'.It describes the number of cubs born each time -the first time five are born, then the second (next) time four, and so on (down) until the fifth time (to the end) only one is born.Next here refers to pregnancy of next time (unit of measure) by the same lioness (subject under discussion).This phrase refers to a time sequence.
21.θ' is not translated, following the practice as in Psalm 119:65, where the alphabet θ' is also not translated.

Testament of Judah 25:1 22
Judah describes what will happen after all twelve sons of Jacob have been resurrected R.H. Charles (1913, 2:324) translates this verse as follows: And after these things shall Abraham and Isaac and Jacob arise unto life, and I and my brethren shall be chief of the tribes of Israel: Levi first, I the second, Joseph third, Benjamin fourth, Simeon fifth, Issachar sixth, and so all in order (πάντες καθεξῆς).
The order here is not time because it does not follow the birth order of the patriarchs in the Bible.It seems that καθεξῆς here describes the order of the twelve patriarchs according to how blessed (by God) they are.The most blessed one is described first, and then the second in order (next) is described, and so on until the least blessed is described (down to the end).Next here refers to the patriarch who is next most important (unit of measure) in the view of God (subject of discussion).The order is a sequence of hierarchy of significance indicating the priority of importance of the patriarchs.

Apocalypsis Moses 8:2 23
Here God tells Adam, after he has eaten from the forbidden tree in the Garden of Eden, that God will bring 72 strokes upon his body.The author then describes the first (about the eye) and the second (about the ear) strokes and then writes 'and likewise in turn (καθεξῆς) all the strokes shall befall thee'.This article translates καθεξῆς as successively.The context seems to indicate that God will impose the strokes one by one upon Adam's body, and if so, καθεξῆς indicates time sequence or chronological order.

The martyrdom of Polycarp 22:4 24
Included is a final comment by the author, which Roberts and Donaldson (1885-1896:n.p.) translate as follows: 'And I again, Pionius, wrote them from the previously written copy, having carefully searched into them ... even as I shall show in what follows ...' (ἐν τῷ καθεξῆς).Holmes (1999:245) translates the Greek phrase as, 'as I will explain in the sequel'.Kαθεξῆς here is used as a noun to describe another book, probably the sequel to the current book.Next here likely refers to the next book (unit of measure) in a spatial sequence written by the same writer (subject under discussion).Kαθεξῆς refers to a spatial sequence concerning the next book the author will write.
1 Clement 37:3 Clement (ed.Lake 1912Lake -1913:n.p.) :n.p.) encourages fellow Christians to follow the command of the Lord.Roberts andDonaldson (1885-1896:n.p.) translate it as follows: Let us then, men and brethren, with all energy act the part of soldiers, in accordance with His holy commandments.Let us consider those who serve under our generals, with what order, obedience, and submissiveness they perform the things which 22.For the Greek sentence with καθεξῆς, see Charles (1966:103).
http://www.indieskriflig.org.zaOpen Access are commanded them.All are not prefects, nor commanders of a thousand, nor of a hundred, nor of fifty, nor the like (τὸ καθεξῆς), but each one in his own rank performs the things commanded by the king and the generals.Holmes (1999:71) translates the phrase in Greek as 'Not all are prefects or tribunes or centurions or captains of fifty and so forth.'My own (Fung's) translation is: 'not all are commanders, nor captains over a thousand, nor centurions, nor leaders of a company of fifty, nor those in (the) order commanding fewer people'.Here this article treats τὸ as a collective noun and καθεξῆς as an adjective modifying it.The context describes an order beginning from the commander who commands the greatest number of soldiers down to the leaders who command the smallest number.Kαθεξῆς is used to describe those who command even fewer soldiers but are not explicitly mentioned by Clement.It seems that if they are mentioned, they appear to be mentioned in the order of decreasing number of soldiers commanded.Kαθεξῆς's meaning 'next down to the end' also fits here where next refers to the leader who commands the next smaller number of soldiers (unit of measure) in the same army (subject under discussion).Kαθεξῆς describes a sequence of hierarchy of significance of decreasing number of people commanded.

References cited in TLG
A search of TLG lists four usages of καθεξῆς between the last century bce and ce first century. 25 Antiochus Astrol., Fragmenta (e cod.Monac.7, 7:114, lines 4 to 14) In this document the context seems to describe an ancient chronological system: And the investigation of the one who acts and who turns about is this: whenever the appointed time of Cronus (Father of Zeus) may become the day, we say; it turns about the whole day and acts the first hour and while it is turning about and acting, it gives the second hour to Zeus and we say the second hour of Cronus … Likewise we say the fourth hour of the Sun, the fifth of the Aphrodite, the sixth of the Hermes and the seventh of the Moon, we say when Cronus is turning about, after appointed time of the day also becomes the one who acts.Then again the eighth hour turns about and acts, and the sequence of the future is thus in order, indeed the sun turns about the first day and acts the first hour, and the moon the second (hour) likewise … Kαθεξῆς is used here to describe time.Although in the passage the way time proceeds is seemingly described in chronological order, this does not necessarily imply that the sequence of the future is also in chronological order, as the sequence of the future is not clearly explained in the context.But if the sequence of the future is the same sequence as described earlier, καθεξῆς probably refers to chronological order.
Onasander Tact., Strategicus (30:T) Here the context seems to describe how to choose a commander for an army: 25.As many usages in the TLG have no English translation, I (Fung) will cite my own translation.
As it is necessary to conclude by way of syllogism the commander before the battle, why ought he to meet during the encounter with the certain one and why with the other and thus one after the other to examine closely his own captains towards the ones of the opponents?
In this passage καθεξῆς is used to describe the method of how to interview people (i.e.'one after the other', or 'one by one').
The passage seems to indicate a kind of logical sequence, though the context does not explain in detail what this sequence is.Kαθεξῆς's meaning, 'next down to the end' also fits here -if the interview is done, the eligible ones will be interviewed according to a logical order until the last one is interviewed.It likely implies a sequential hierarchy of significance.
Vitae Aesopi, Vita Aesopi Westermanniana (40, lines 7-8) Here is a description of a conversation between a master and a subordinate: You have commanded me to hear everything, and to do those things.You did not say: Throw water into the pot and wash my feet and place the sandals and everything in their appropriate place.
Kαθεξῆς describes 'the appropriate locations for the subjects which should be placed'.It indicates some kind of spatial sequence.Kαθεξῆς's meaning 'next down to the end' fits herethe items will be placed one by one according to a spatial order until the last one is placed, though the type of spatial order is not specified.
Heron's Geodaesia [Sp.] (6:2, lines 1-5) It describes sowing.It is necessary to know that a peck measure fit for sowing contains 40 litra, and every one litra sows a land of 5 fathoms.For width and length of 5 fathoms while doing 1 litre, also in (the following) order: Width and length of 10 fathoms while doing 2 litres Width and length of 15 fathoms while doing 3 litres Width and length of 20 fathoms while doing 4 litres… Kαθεξῆς describes the area of land which can be sown with the increasing number of litres used.Kαθεξῆς's meaning 'next down to the end' is appropriate here -the description continues through the maximum number of litres that can be used.Next here refers to the next additional litre (unit of measure) used for the extra area of land sown (subject of discussion).Kαθεξῆς here is categorised as a spatial sequence.
In summary, the study of καθεξῆς in contemporary Greek usages indicates that it does not usually refer to just any logical order.Plutarch's Moralia shows that the way the myrtle branch passes along is not considered as καθεξῆς.Kαθεξῆς likely means 'next down to the end' with next being the neighbour closest to the one before in the unit of measure explicitly mentioned or implied, and it shares the same subject under discussion with the one before.Of the 16 references of καθεξῆς studied, 26 7 (43.8%)refer to time sequence, 5 (31.2%) spatial sequence, and 4 (25%) hierarchy of significance.Time sequence is the most common meaning of καθεξῆς, and when it refers to time sequence, it always refers to chronological order.

Conclusion
There are several observations about the potential meaning of καθεξῆς which seem to support the conclusion that καθεξῆς in Luke 1:3 most likely refers to 'chronological order'.They are as follows: Based on a study of the etymology of καθεξῆς, the word likely has a meaning of 'next down to the end' or 'next in order to, down to the end'.Therefore, if this is applied to Luke 1:3, καθεξῆς probably means that Luke writes one event after another according to their time of happening ('next in order to') for everything he has investigated ('down to the end')signifying that Luke most likely writes in chronological order.
Based on a study of contemporary Greek usages of καθεξῆς, καθεξῆς does not refer to just any logical order.Kαθεξῆς refers to a sequence and the next in its 'next down to the end' definition refers to the neighbour closest to the predecessor in the unit of measure explicitly mentioned or implied in the context, and it shares the same subject under discussion with the predecessor.Most references are to time, and secondarily to space.The least frequent references to logic are references to hierarchy of significance.
If Luke's understanding of the potential of meaning of καθεξῆς is the same as that of his contemporaries, καθεξῆς likely refers to a sequence which has the meaning 'next down to the end', with next referring to the next act or teaching closest in time (the unit of measure) to the one previously conducted by Jesus (subject under discussion), implying chronological order.
Besides Luke 1:3, καθεξῆς occurs four times only in Luke and Acts in the NT, and the usages in Luke 8:1, Acts 3:24 and 11:4 (i.e.75%) refer to time sequence.Root components of καθεξῆς are also helpful in understanding the word.The roots κατά and ἑξῆς literally have to do with 'next down to the end' or 'next in order to, and down to the end'.A sequence or order seems to be the basic idea of the root ἑξῆς.In this research, the author (Fung)  in the Gospel.Nevertheless, Luke himself used ἑξῆς only with time.In addition, since καθεξῆς is used of a whole gospel, space or physical proximity seems not to be a relevant category.The biblical references would argue more for chronology, while the contemporary Greek references would prefer chronology, as well, though hierarchy of significance is possible.Some scholars have suggested that καθεξῆς has to do with thematic logic.However, in the evidence of this study, thematic sequencing does not appear to be an aspect of καθεξῆς.The study shows that BDAG (2000:490) is correct in referring καθεξῆς first to time; καθεξῆς also refers to space or logic (i.e.hierarchy of significance), but those categories are less frequent.
some time he went out while passing through in order the Galatian country and Phrygia, strengthening all the disciples' (διερχόμενος καθεξῆς).Later in his journey Paul reaches Ephesus (19:1).According to the geographic locations of Antioch, Galatia, Phrygia and Ephesus, Antioch is east of Galatia, Galatia east of Phrygia, and Phrygia east of Ephesus.It seems that Paul is gradually traveling west away from Antioch and towards Ephesus.If this is true, καθεξῆς here means 'in the order from the cities which are closer in distance west of Antioch (i.e.those cities in the Galatian country) to the ones further away west of it (i.e.those in Phrygia)', which appears to be an order of increasing geographical distance.Therefore, this article categorises it as spatial.
5.All translations of the Greek are the author B.W.W. Fung's own literal version.ESV and NRS translate this phrase as 'soon afterwards', NET as 'some time afterwards', and REB and NIV as 'after this'.6.NRS translates τῶν καθεξῆς as 'those after him', ESV as 'those who came after him', NET as 'those who followed him', NIV as '(from Samuel) on', and REB as 'From Samuel onwards'.7.ESV translates καθεξῆς as 'in order', NRSV as 'step by step', NET as 'point by point', NIV as 'everything … precisely as it had happened', and REB as 'the facts as they had happened'.spending , it means 'next in a series' or 'in the next place', and according to LSJ  (1996:594)the more important meanings of the word are 'next', if used to indicate time; 'next to' or 'next in order to', if used to indicate logical connection; and 'next place', if used to indicate place.It is used only five times in the NT,Macc 1:9).This article studies the meaning of ἑξῆς in each of the above 11 occurrences and categorises them according to LSJ's three major categories mentioned above.10Luke7:11 describes Jesus and his apostles going to a town called Nain.In the context ἐγένετο ἐν τῷ ἑξῆς means literally: 'it happened in the next'.As no noun comes after ἑξῆς, it is not Similar to the logic mentioned above, this article translates the phrase as 'in the next (day) into Rhodes and into Patara' as an implied ἡμέρᾳ in the feminine matches the feminine τῇ.The phrase is similar to τῇ ἑξῆς ἡμέρᾳ used next (day) (τῇ ἑξῆς) we were throwing cargo overboard'.Again, Luke appears to use the feminine τῇ ἑξῆς to describe time and to imply the 'next' day.In summary, in all five usages of ἑξῆς, Luke does not use the word separately but always with a dative article (either τῷ or τῇ).He uses ἑξῆς to describe time sequence which means either 'the next time' or 'the next day'.Luke seems to understand ἑξῆς, when used to describe time sequence, as meaning next.This agrees with LSJ's observation that if ἑξῆς is used to indicate time, it means 'next'.
Fung (2017)er Luke wants to imply χρόνῳ (time -i.e. next time or soon afterwards) or ἡμέρᾳ (day -i.e. next day) as the noun in this phrase, and Luke mentions neither χρόνῳ nor ἡμέρᾳ earlier.BDAG (2000:349)concludes that it should be time and reads 'soon afterwards', while LSJ (1996:594)suggests that it should be 'day' and reads 'in the next day'.Since τῷ is in the masculine or neuter form which matches the masculine form of χρόνῳ, and ἡμέρᾳ is in the feminine form,Fung (2017)prefers χρόνῳ to ἡμέρᾳ.Therefore, this article translates this verse as 'And it happened soon afterwards He went into a city called Nain and his disciples and a great crowd went with Him.' Based on the context, here the word ἑξῆς means next and is related to time.At the beginning of Luke 9:37 Luke uses ἑξῆς in a phrase which is very similar to the phrase in 7:11.Luke 9:37 says that the next day, when Jesus and the apostles come down from the mountain, a great crowd meets Him.Ἐγένετο δὲ τῇ ἑξῆς ἡμέρᾳ may be translated as 'and it happened in the next day'.Again ἑξῆς (next) is related to time.The meaning is similar to the phrase ἐγένετο ἐν τῷ ἑξῆς ('it happened in the next time' or 'soon afterwards') used in 7:11 which shares the same construction.In Acts 21:1b Luke is describing one of Paul's missionary trips: τῇ δὲ ἑξῆς εἰς τὴν Ῥόδον κἀκεῖθεν εἰς Πάταρα.Again Luke does not supply the noun for τῇ … ἑξῆς.thejudgment seat, I ordered the man to be brought in.'Here τῇ ἑξῆς is the same as in Luke 9:37 and Acts 21:1, and the context seems to support ἑξῆς being related to time and means 'next'.Acts 27:18 describes the ship Paul is on, facing a big storm: 'and while we were being violently tossed in 10.Refer to Appendix 2 for the meanings of ἑξῆς.the In summary, considering the meanings of its root words, κατά and ἑξῆς or ἔχω, καθεξῆς probably has the meaning of next or next in order to (from ἔχω or ἑξῆς) and down (from κατά).With the perfective force provided by κατά, καθεξῆς likely means next down to the end or next in order to, and down to the end.Similarly, in Luke 1:3 Luke intends to write one event after another according to their time of happening (next, or next in order to and down) for everything he has investigated (to the end).By using καθεξῆς, Luke suggests that he writes his Gospel in chronological order.
11ἀνοχή 12 and related phrases13do not provide any helpful information as to the meaning of καθεξῆς.They pick up the meaning of to have from ἔχω instead of next.LSJ  (1996:852) and BDAG (2000:489)do list ἐφεξῆς, which can be used interchangeably with καθεξῆς.It means in order, one after another, in a row, and next to, successively, continuously.If used to describe time (e.g.τρεῖς ἡμέρας ἐπεξῆς meaning three successive days), it means thereupon and immediately afterwards.14Middle and Liddell in the TLG database also define this word as 'in succession, without exception, of time'.Its meanings of successively, continuously and in succession, without exception, of time seem to support the idea that καθεξῆς can mean in chronological order.
http://www.indieskriflig.org.zaOpen Access κάθεξις, These include Plutarch's Moralia, finds sequencing that is organised by time, space or hierarchy of importance. 27Nevertheless, Luke always uses ἑξῆς to describe a time sequence, whether referring generally to time or specifically to hour.In the Septuagint ἑξῆς is used for time, space, and hierarchy of significance.Kαθεξῆς does occur outside the Bible in contemporary Greek examples, where it is used to describe sequence of time, space (physical proximity) and hierarchy of significance.Thus, any of these possibilities can occur 26.See Appendix 1.27.See Appendix 2.