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Note: You may wish to read Chiastic Structuring: An Introduction first.
That Matthew's Sermon on the Mount has a chiastic structure has been argued by Austin Farrer  and accepted by J. C. Fenton.  Farrer's suggested structure arranged the body of the Sermon chiastically against the first eight beatitudes (Matt 5.3-10). We shall argue, on the contrary, that the nine beatitudes (5.3-12) are arranged in groups of three, and that 5.13-7.11 is chiastic against them.
Furthermore, we intend to demonstrate that the whole of the Sermon, 5.1-7.29, is chiastic against 22.15-24.35, which includes the woes of chapter 23. Thus 5.1-7.29; 22.15-24.35 appears to have been a unified pre-Matthaean unit. It probably grew out of something very like the brief beatitudes and woes of Luke 6.20-26 to constitute a type of Two-Ways teaching unit not too dissimilar to those found in Barn. 18.1-21.9 (the way of "light" versus that of "darkness") and Did. 1.1-6.2 (the way of "life" versus that of "death").
The discrete endings of the two halves of this lengthy chiasm indicate that it has been taken up by the evangelist and deliberately separated into its two halves within the larger structure of his gospel, with the woes materials now directly aimed against the "enemies" of the Matthaean community, namely the scribes and Pharisees.
The following presentation is in three parts. The first is concerned with explaining the structure and import of the nine beatitudes of Matt 5.3-12 as constituting three groups of three each. The second relates Matt 5.3-12 chiastically to Matt 5.13-7.11. The third relates Matt 5.1-7.28 chiastically to Matt 22.15-24.34.
Matt 5.3-12: The Beatitudes
Before we look at the beatitudes themselves some background on the structure of the whole of Matthew is in order, based on work published elsewhere, but which is fully presented in Wisdom, Power and Well-being: Ancient (Biblical) Parameters for Humanity. From no later than the eighth century BCE as found in Isaiah (e.g. Isa 3.1-3) or even possibly the tenth century as found in the Joseph cycle (Gen 41.39-45)  a triadic pattern of wisdom, power and wellbeing (or riches) was used among the Israelites to define, positively or negatively, the humanity of individuals, groups or nations. When used positively, they defined the humanity of those who live in total dependence upon God for these parameters. When used negatively they define the (in-)humanity of those who set themselves over against God.
In Matthew the two key passages for understanding the evangelist's structuring are 1.1 and 23.23. Matt 1.1, which is recognized as being a re-writing of Gen 5.1 (LXX), substituting "Jesus" for "Adam", reads: "The book of the generations of Jesus, Christ, Son of David, Son of Abraham." Matt 23.23 speaks of "the deep things of the law: justice, mercy and faith." As the Christ Jesus is the servant who brings justice to victory, to completion, in and through the deed of the cross by total dependence upon God (cf. Matt 12.18-21 citing Isa 42.1-4). As Son of David he is the one who is called upon for, and shows, mercy,  the wisdom of God. As Son of Abraham he is the one who totally depends upon the Father for his wellbeing, living by his every word. This pattern of wellbeing, wisdom and power, in that order, structures Matthew's overall narrative as well as lesser units such as the temptation narrative (4.1-11) and the Lord's Prayer (6.6-13) as it also does the Beatitudes and the main body of the Sermon on the Mount, as we shall show.
As we examine the beatitudes themselves, we shall see that the middle beatitude in each set is flanked by a pair which appear to correspond to each other.
The first set, 5.3-5, concern the poor in spirit, those who mourn, and the meek.
The poor in spirit appear to correspond to the meek, and the kingdom of heaven that is allotted to the former is balanced by the inheriting of the earth allotted to the latter, both of which belong to Jesus (28.18). Here in 5.3-5 is a further link with Abraham who was held to have acquired both worlds by faith.  The verse concerning the meek (5.5) cites Ps 37.11. Ps 37.10 reads: "For yet a little while and the wicked shall not be: Yea, thou shalt diligently consider his place, and he shall not be." This increases the likelihood that the meek are those who will endure, those who will live and have wellbeing.
If the poor in spirit and the meek can readily be seen to be those who live by faith and not presumption, then in what way do those who mourn (5.4) fit this same theme? We would suggest that this is to be connected with sons of Abraham in terms of the relationship of the following passages: 3.6-9; 5.4; 9.15 and 11.18. In 5.4 the verb for "mourn" is πενθεῖν, which otherwise occurs only in 9.15. In 9.15, answering the question by John's disciples as to why Jesus' disciples do not fast (9.14), Jesus replies, "Can the sons of the bridechamber mourn as long as the bridegroom is with them? But the days will come, when the bridegroom shall be taken away from them, and then they will fast." Thus in Jesus' reply fasting is presented as a form of mourning. The verb πενθεῖν is associated with mourning for sin, either a sorrow for one's own sin (Test. Reub. 1.10; 1 Cor 5.2), or more often a sorrow for the sins of others (1 Esdr 8.69; 9.2; 2 Esdr 10.6).  In Matt 11.18 Jesus says, "John came neither eating nor drinking," which is balanced chiastically  against "We wailed and you did not beat your breast" (11.17), that is, you did not mourn.  Again, as in 9.15, we see the conjunction of John, fasting and mourning, a mourning of repentance. In Matt 3.1 John comes preaching repentance; in 3.6 those baptized confess their sins; in 3.8-9 John warns the Pharisees and Sadducees, "Bring forth therefore fruit worthy of repentance, and think not to say to yourselves, 'We have Abraham to our father,' for I say to you that God is able from these stones to raise up children to Abraham." Thus those who mourn in the beatitude of 5.4 would appear to be those who truly repent and are allied with the wellborn children whom God raises up to Abraham. We may therefore conclude that the first three beatitudes, 5.3-5, are about those who live by faith, the wellborn children of Abraham, who claim nothing of themselves. 
The second set, 6.6-8 concerns those who hunger and thirst after righteousness, those who are merciful, and those who are pure in heart.
The fourth one, 5.6, brings to mind the later passage in the Sermon, 6.31-33, in which, instead of asking, "What shall we eat? or, what shall we drink?", for these needs are already known to "your Father" (6.32), the disciples should instead, "Seek first his kingdom and his righteousness, and all these things will be added unto you" (6.33).
In the sixth one, 5.8, the pure in heart means those who are single-minded, who desire only to be under God's reign and his righteousness. In Ps 24.4-6 it is said in the Greek and Syriac versions that the poor in heart seek the face of the God of Jacob (MT: "that seek thy face, O Jacob"), while Ps 11.7 and 17.15 speak of the righteous as those who shall see the face of God. Thus the fourth and sixth beatitudes, 5.6, 8, appear very likely to be equivalents.
Since the fifth one speaks of the merciful, we have only to show that the fourth and sixth ones also dominantly concern mercy. That Matthew is concerned with Torah can readily be seen from such passages as 5.17-18; 7.12; 11.13 and 22.36-40, in all of which Jesus speaks of the Law and the Prophets. The issue between Jesus and the Pharisees is presented as being one of teaching the proper interpretation of Torah. This is seen in such passages as 16.12 (Disciples are to beware of the teaching of the Pharisees and Sadducees) and 15.12 (disciples report that the Pharisees are scandalized by Jesus' teaching, and Jesus replies that the Pharisees are blind guides, 15.14). The issue with the Pharisees is over the central content of the Torah. Three times in Matthew Jesus confronts the Pharisees over the issue of mercy . Besides 23.23, where justice, mercy and faith are the deep things of the Torah that the scribes and Pharisees are said to have left undone but ought to have done, we also have 9.13 and 12.7. In 9.13, quoting Hos 6.6, Jesus says to the Pharisees, "But go and learn what this means: 'I desire mercy and not sacrifice.'" In 12.7 he again quotes the same passage to the Pharisees, "But if you had known what this means: 'I desire mercy and not sacrifice,' you would not have condemned the guiltless." Twice Matthew, and Matthew alone among NT writers, cites the Hosea passage, and both times the Pharisees are accused of not knowing what it means. We would suggest strongly, in view of the correspondence of the Matthaean note of "mercy" to the Pauline note of "love," that "mercy" is being presented in Matthew as the greatest of the three notes of justice, mercy and faith, just as "love" is the greatest of the three enduring elements of faith, hope and love in 1 Cor 13.13.  On this basis we would see the righteousness of Matt 5.6 and the pureness of heart of 5.8 as centering on mercy, the present grace and demand of God.  If this be the case, then our second set of beatitudes. 5.6-8, is centered on the theme of mercy, the attribute ascribed in Matthew to Jesus as the Son of David. In any case, that these three beatitudes pertain to David can be readily seen in the "Song of David", 2 Sam 22.25-27a:
Therefore has Yahweh recompensed me according to my
righteousness, according to my cleanness in his sight.
With the merciful thou dost show thyself merciful,
with the blameless man thou dost show thyself blameless;
with the pure thou dost show thyself pure ....
The third set, 5.9-12, concern peacemakers, those persecuted for righteousness' sake, and those being addressed by Jesus when they are reviled and persecuted for his sake.
That the first and third, 5.9, 11-12, are related is indicated by 5.44-45: "Love your enemies and pray for those who persecute you, that you may be sons of your heavenly Father." Here is peacemaking corresponding to 5.9a, and here is the title "sons of God" which is what the peacemakers are to be called, 5.9b. Here also is persecution as in 5.11, and 5.46 goes on to mention the matter of reward: "For if you love those that love you, what reward have you?" This matches the reward of 5.12: "Great is your reward in heaven, for so persecuted they the prophets who were before you." Matt 23.29-35 indicates that persecution and crucifixion at the hands of the scribes and Pharisees is to be the lot of those whom Jesus will send, as it was the lot of the prophets. Thus we would see the seventh (5.9) and ninth (5.11-12) beatitudes as related not only to each other, but as integrally related also to the eighth (5.10). Obviously, the way of the cross comes to mind, and it is as "that righteous one" (27.19) that Jesus pours out his blood for the forgiveness of sins (26.28; cf. 27.24 and 23.35) as the Christ (26.63, 68; 27.17, 22).
In summary form then we have:
|Wellbeing||Son of Abraham||Faith||4.3-4||5.3-6|
|Wisdom||Son of David||Mercy
Relating the Beatitudes to the body of the Sermon
We shall now relate 5.13-7.11 chiastically to the beatitudes of 5.3-12.
|Beatitudes||Body of Sermon|
|5.11-12||Blessed are you||5.13-16||You are the salt of the earth|
|5.10||persecuted for righteousness' sake ... kingdom||5.17-20||your
enter kingdom of heaven (5.20)
|5.9||peacemakers||5.21-26||be reconciled to thy brother (5.24)|
|5.8||Pure in heart||5.27-37||Adultery in his heart (5.28); offending eye (5.29) or heart (5.30); let your yes be yes (5.37)|
|5.7||The merciful||5.38-43||Love your enemies (5.44)|
|5.6||Hunger and thirst after righteousness||6.1-18||Your righteousness (6.1); the three Pharisaic works of almsgiving (6.2-4), prayer (6.6-15) and fasting (6.16-18)|
|5.5||The meek ... inherit the earth||6.19-34||Lay not up treasures on earth (6.19) but in heaven (6.20); seek Father's kingdom and his righteousness, and all these things shall be added to you (6.33)|
|5.4||Mourners [for sin] shall be comforted||7.1-5||Judge not that you be not judged (7.1); the mote and the beam (7.3-5) - [i.e. repent for your own sins, not for others' sins]|
|5.3||Poor in spirit - theirs is the kingdom of heaven||7.6-11||Give not the holy things to dogs (7.6); ask, seek, knock (7.7-8) - [i.e. as one in need, not as one claiming self-sufficiency]; your heavenly Father gives good things to those who ask (7.11) |
The correlation ends at 7.11. It is followed by the summary statement from Q (cf. Luke 6.31), the so-called Golden Rule, which Matthew closes with the words, 'This is the Law and the Prophets'. The remainder of the sermon (7.13-27) consists of appended words of warning about the narrow gate that leads to life (7.13-14), and the need to hear and do Jesus' words (7.24-27). similar types of warning materials are to be found in all of the Matthaean discourses, invariably coming at the end if they are in the form of parables.
The Chiasm of 5.1-7.29 against 22.15-24.35
|5.1-2||He opened his mouth and taught them||24.32-34||I say to you ... my words shall not pass away.|
|.3||Poor in spirit - kingdom of heaven will be theirs||.31||Gather his elect from one end of heaven to the other|
|.4||Mourn ... be comforted||.30a||will mourn; see Son of Man come power/glory|
|.5||Meek ... inherit earth||.29-30a||Tribes of earth|
|.6||Hunger and thirst for righteousness - be satisfied||.23-28||Seeking Christ|
|.7||Merciful obtain mercy||.16-22||For sake of elect days shortened|
|.8||Pure in heart see  God .||.15||See desolating sacrilege |
|.9||Peacemakers - sons of God|
|.10||Persecuted for righteousness' sake - kingdom of heaven||.13-14||Endure to end - be saved. This gospel of the kingdom|
|.11||Persecuted, evil against you on my account||.9-12||Hated by all nations for my name's sake|
|.12||Reward great in heaven||.3-8||Take heed no one leads you astray|
|.13||Salt of earth - if saltless, thrown out (βληθὲν ἔξω)||.1-2||Stones of temple shall be thrown down (καταλυθήσεται)|
|.14-16||Light of world, city on hill; by good works give glory to Father||23 .37-39||O Jerusalem, Jerusalem [Jesus' cry as Wisdom ; cf. 11.19 - justified by her works]|
have come to fulfil
Law and Prophets
measure of your fathers - brood of vipers;
34: I send prophets, wise ones, scribes [Jesus speaking as Wisdom - cp. Luke 11.49]
|.18||until all things come to pass||.36||All this will come upon this generation|
|.21-26||Do not kill - gift at altar - be reconciled; make friends with accuser.||.27-28||Full of dead men's bones - full of hypocrisy and iniquity|
|.27-30||No adultery; lust - adultery in heart; body into Gehenna||.25-26||Full of extortion and rapacity|
|.31-32||Certificate of divorce - adultery||.23-24||Neglect depths of Torah; strain gnat, swallow camel|
|.33-37||No swearing||.16-22||Forswearing |
|.38-42||Turn other cheek; go second mile||.13-15||You shut kingdom of heaven against men|
|.39||Resist not him who is evil||.15||Sons of Gehenna|
|.43-44||Love enemies||.12||Greatest be servant of all|
|.45||as your Father in heaven loves||.10-11||[as] Christ your Guide [served]|
|.48-6.1||Be perfect as your Father in heaven||.8b||You have one Teacher|
|6.2a||Alms: no trumpet in||.7||market places|
|.2b||synagogues and streets||.6||synagogues|
|.2c||to be praised by men||.5a||to be seen by men|
|.3-4||Not let left hand know what right hand does; Father will reward you.|
|.5-15||Prayer||.5bc||Phylacteries and fringes|
|.16-18||Fasting||.6||Chief seats at feasts|
|.19-32||Anxiety||.4||Heavy burdens, grievous to be borne|
|.33-34||Seek first Father's kingdom and righteousness|
|7.1-5||Receive measure meted; do||.3b||They say; they do not|
|.6||Holy things to dogs; pearls to swine||.1-3a||Do what they say; they sit in Moses' seat|
|.7-11||Your Father in heaven - good things to son||22.36-41||Whose Father is Christ's? (David?)|
|.12||The Law and the Prophets: As you would have others do to you, do so to them||.34-40||Basis of the Law and the Prophets: Love God and neighbour as you love yourself|
|.13||Gate to life||.23-33||[Sadducees] v.32: God of the living; vv.28, 30: resurrection|
|.15-27||False prophets and their fall||.18-22||Pharisees' disciples and Herodians|
|.28-29||The crowds astonished at his teaching; he taught as having authority, not as scribes||.15-17||Teacher, you teach the way of God in truth; tell us, is it lawful...?|
1 Austin Farrer, St Matthew and St Mark (London, 1954) chapter 10. (return to text)
2 J. C. Fenton, "Inclusio and Chiasmus in Matthew," Studia Evangelica Ι edited by F. L. Cross (TU 73; Berlin: Akademie-Verlag, 1959) 178. (return to text)
3 H. D. Betz, noting the wide gate that leads to life and the narrow gate that leads to destruction of Matt 7.13-14, recognizes the motif of the Two Ways. He rightly says that "the S[ermon on the] M[ount] in its entirety is be regarded as 'the way to eternal life,' whereas 'the way to destruction' consists of the doctrines and practices explicitly or implicitly rejected by the S[ermon on the] M[ount]" (Essays on the Sermon on the Mount [Philadelphia: Fortress Press, 1985] 3). While arguing that the Sermon on the Mount is "a pre-Matthaean source composed by a redactor" (ibid. x), he concludes that the Sermon on the Mount is not of the Two Ways genre because he has not discerned the pre-Matthaean structural relationship of 5.1-7.29 to 22.15-24.35 which balances the Sermon on the Mount (the way to life) against the Woes (the way to destruction). Betz repeats this view in his Hermenia commentary on the Matthaean and Lukan sermons. (return to text)
4 James M. Gibbs, "Wisdom, Power and Wellbeing: A Set of Biblical Parameters for Man and their Use in the New Testament to undergird Jesus' and the Christian's Humanity" Studia Biblica 1978 III edited by E. A. Livingstone (JSNT Suppl. Series, 3, Sheffield, 1980) 119-155. (return to text)
5 If the Joseph cycle has a largely Egyptian setting, then it is perhaps worth noting that in Memphis, the capital in the north during the New Kingdom (c. 1539-1075 BCE) in Egypt, three deities were associated together in terms of funerary rites: Ptah, creator god of wisdom, imagination and craft; Sakhmet, fierce protectress, and their child, Nefertem, god of fertility and new life (this information was presented in "The Quest for Immortality: Treasures of Ancient Egypt" Exhibition at the National Gallery of Art, Washingon, D.C., 2002). On the basis of the work of Georges Dumézil on Indo-European languages, it would appear that the origin of the wise - powerful - wellborn pattern probably lay in the division of society into rulers/sages (wisdom), warriors (power) and hunter-gatherers (suppliers of wellbeing). (return to text)
6 See James M. Gibbs, "Pattern and Purpose in Matthew's Use of the Title 'Son of David,'" NTS 10 (1964) 446-464. (return to text)
7 See the entries under "Abraham" in the index of G. F. Moore, Judaism Vol. 2 (Cambridge, Mass., 1946) 399. (return to text)
8 BAGD,.πενθέω (return to text)
9 See J. C. Fenton, "Inclusio and Chiasmus in Matthew", 176. (return to text)
10 This middle voice of κόπτειν, which means striking one's breast in mourning, occurs elsewhere in Matthew only in 24.30 concerning the mourning of all tribes of the earth when the Son of man appears. (return to text)
11 This helps to make sense of Jesus' submission to John's baptism of repentance as part of his fulfilling all righteousness (3.15). Perhaps Jesus' fasting forty days and forty nights in the wilderness (4.1-2) is to be seen in part as his mourning as the righteous remnant of Israel, the remnant of One, for Israel's forty years of waywardness in the wilderness. That he represents Israel in its calling as Son of God (Exod 4.22-23) is shown by the flight into Egypt for safety (Matt 2.13-14, a Jacob/Israel typology, going to Egypt for safety, as opposed to a Moses typology which would involve a fleeing from Egypt) which is followed by the quoting of Hosea 11.1 in Matt 2.15: "Out of Egypt did I call my Son." If this is to be seen as such a fast, then it is only after it (4.2) that he successfully overcomes the temptations to which Israel fell prey. (return to text)
12 In Wis 7.22b-8.8 Wisdom herself is said to encompass wisdom (7.22b-23; 8.4a, 7-8), power (7.24-25, 27-28, 30-8.1; 8.4b, 5c-6) and wellbeing (7.26, 29; 8.2-3, 5). On this basis wisdom as "love" in 1 Corinthians and as "mercy" in Matthew may well be the basis of and encompass the other two, namely, "faith" and "hope" in Paul and "faith" and "justice" in Matthew. (return to text)
13 Luke 6.36 has "be merciful as your Father is merciful," which concentrates the demand even more. But the Matthaean form, "Be perfect as your heavenly Father is perfect" (5.48), may be intended to cover all three of the deep things of Torah. (return to text)
14 The loaf and stone of Matt 7.9 suggest the first temptation, 4.3-4. (return to text)
15 In John 14.9 Jesus says, "He who has seen me has seen the Father," and the Greeks' request, "Sir, we would see Jesus" (John 12.21), is immediately followed by pointing forward to the passion, so that the true seeing of Jesus is at the cross. In similar vein in Matthew the seeing of Jesus on the cross elicits the confession, "Truly God's Son was this one" (Matt 27.54), and this is where he is seen as God-with-us (1.23). (return to text)
16 Note that in Mark 14.63-64 and Matt 26.65 (but not in Luke 22.71) the high priest rends his garments in two at "blasphemy"; then God rends the temple veil in two at the desolating sacrilege/blasphemy of the cross, emphatically stated in Mark 15.38 and Matt 27.51, but greatly toned down in Luke 23.45. We would argue that Luke has removed the charge and counter-charge of blasphemy, including the desolating sacrilege (cp. Luke 21.10 to Mark 13.14 and Matt 24.15) as part of an attempt to rehabilitate the appeal of the gospel to the Jews. (return to text)
17 That Jesus himself is Wisdom in Matthew is made clear by the terms of 11.25-30. James M. Gibbs, "The Son of God as the Torah Incarnate in Matthew", Studia Evangelica IV ed. F. L. Cross (Akademie-Verlag, Berlin, 1968) 38-46. (return to text)
18 This section on the voiding of oaths made to others (23.16-22) repeatedly refers to the temple. We have seen that David, the figure of mercy, is connected with the temple. If the chiastic structure proposed here is accepted, then the corresponding section, 5.33-37, which we have presented as being concerned with one of the beatitudes about mercy, is even more likely to be so on the basis of the temple motif here in 23.16-22. The relation of 23.16-22 to the second temptation with its temple motif (4.5) is quite evident. (return to text)