AND THE LAW CAME IN THAT THE TRANSGRESSION MIGHT INCREASE: nomos de pareiselthen (3SAAI)  hina pleonase (3SAAS) to paraptoma: (Ro 3:19,20; 4:15; 6:14; 7:5-13; Jn 15:22; 2 Cor 3:7-9; Gal 3:19-25)

Denny introduces these last 2 verses of Romans 5 commenting that...

The comparison between Adam and Christ is closed. But in the middle, between the two, stood the Law. (quoting Meyer)

Adam may have faded from the discussion but the consequences of his one transgression (sin and death) linger on.

Marvin Vincent agrees writing that...

Now that the parallel between Adam and Christ is closed, the question arises as to the position and office of the law. How did it stand related to Adam and Christ? Paul replies that it came in alongside of the sin. “It was taken up into the divine plan or arrangement, and made an occasion for the abounding of grace in the opening of the new way to justification and life” (Dwight). (Vincent, M. R.  Word Studies in the New Testament. Vol. 3, Page 1-65)

And the Law came in - Refers to God's giving of the Law to Moses and Israel at Mount Sinai.

Law (3551) (nomos from nemo = to divide among, parcel out, allot) according to Vine primarily meant “that which is assigned” and hence, that which is the “custom” and finally that which is “law as prescribed by custom or by statute”. The word ethos, “custom,” was retained for unwritten “law,” while nomos became the established name for “law” as decreed by a state and set up as the a formalized set of rules for the administration of justice.

In this context Law refers to the Mosaic Law given at Mount Sinai (it "came in beside" sin and death) (See related topics - Summary of the Purpose of the Law from Galatians 3; see also William Newell's discussion of the purpose of the Law in exposition of Romans 3:20) Nomos can refer to the first five books of Moses (the Pentateuch or the Torah in Hebrew) and that may be Paul's meaning here, but the emphasis is clearly on the legal aspect, in the sense of prescribing what a person must do and not do.

One might ask "Why bring up the Law at this juncture?" Leon Morris explains that...

It is somewhat unexpected that he brings in a reference to law, but between Adam and Christ stood Moses, revered by the Jews and often seen as the most significant figure among the sons of men because of his giving of the law. This was so important in Jewish religion and in the Old Testament that Christians as well as Jews regarded as Scripture that it had to be seen in its proper place in God’s great scheme of salvation. Law, says Paul, was added, the verb (pareiserchomai) showing that it held no primary place. Its purpose was that the trespass (Ed note: trespass singular - see discussion below) might increase. It was not concerned with preventing sin (it was too late for that {Ed note: Adam's original sin}). Nor was it concerned with preventing (Ed note: continual committing of sins) sin (it was too weak for that). The law can only condemn (see note Romans 4:15). It was concerned with showing sin for what it is, and it certainly showed magnificently that there was much sin (cf. notes Romans 3:19; 3:20). (Morris, L. The Epistle to the Romans. W. B. Eerdmans; Inter-Varsity Press)

Came in (3922) (pareiserchomai  from pará = alongside + eisérchomai = to enter) means to come in to beside and in some context means to slip in or to sneak in (see use in Gal 2:4 below). To come in with something so as to be present beside it.

Thayer says the idea here in Romans is "to enter in addition" (in addition to or "beside" sin and death which had previously entered into the Garden of Eden through Adam).

Paul also has the only other use of pareiserchomai in Scripture...

Galatians 2:4 But it was because of the false brethren (Judaizers - claimed allegiance to Christ but demanded circumcision + obedience to the Law for salvation) who had sneaked in (pareiserchomai) to spy out (means primarily to view closely and in this context to spy out, learn about by secret observation) our liberty (eleutheria - see study of related verb eleutheroo = the idea is of freedom and in this context of freedom from the law as a means of salvation and sanctification but not a license to sin!) which we have in Christ Jesus, in order to bring us into bondage (Reduce to slavery, imposing control over another for one's own ends = Total enslavement to a system of works righteousness by which salvation and/or sanctification is impossible). (Comment: In this context the meaning is clearly to come in or slip in secretly, by stealth)

BDAG adds that pareiserchomai means...

to come in beside, slip in, come in as a side issue, of the law, as having no primary place in the divine plan. (Arndt, W., Danker, F. W., & Bauer, W. A Greek-English Lexicon of the New Testament and Other Early Christian Literature)

Friberg says that the idea of this verb is that..

the (Mosaic) law (was) brought in to play a subordinate role (Friberg, T., Friberg, B., & Miller, N. F. Analytical Lexicon of the Greek New Testament. Baker Academic)

Cranfield adds that the prefix para in this verb means...

'alongside of’, ‘beside’, and the most natural way of understanding pareiserchomai here is surely to take it as a simple reference to the undisputed fact that the law was given at a later date than that of Adam’s fall, namely, in the time of Moses. To refer to this fact is not, in itself, to say anything about the worth of the law depreciatory or otherwise. (Cranfield, C. E. B Exegetical Commentary on the Epistle to the Romans. Vol 1: Ro 1-8.; Volume 2: Romans 9-16)

Paul used the root verb eiserchomai earlier in Romans 5:12 when he explained that

through one man sin entered (eiserchomai) into the world (see note Romans 5:12)

So just as sin entered the world through Adam in Romans 5:12, here in Romans 5:20 Paul says the Law of Moses "entered in beside" or alongside sin. Men were sinners long before the Law was given and God had begun implementation of His plan of salvation before the Law was given (eg, see Ge 3:15 - multiple translations; Ephesians 1:4 - note). The purpose of the Law’s entrance into the world was not to redeem men for only Christ could accomplish this great objective. Don't misunderstand the purpose of the Law for it was not given in order to make men sinful. As Paul explained in Romans 5:12-19 man did not need to be made sinful for he is born sinful and that is why he commits sins. Instead, Paul explains that the Law was given so that sin might be made more evident.

Compare this role of the Law with Paul's earlier point in Romans 3:20 [note] where he explained that...

by the works of the Law no flesh will be justified (declared righteous) in His sight; for through the Law comes the knowledge of sin. (See discussion)

In Galatians 3 Paul explained that the promise of God is superior to the Law which raised the obvious question of...

Why the Law then? It was added because of transgressions, having been ordained through angels (see Acts 7:53, Hebrews 2:2) by the agency of a mediator, until the Seed (the Messiah) should come to Whom the promise had been made. (See note Galatians 3:19) (Comment: The Law was intended to reveal sin in its true character as transgression and prepare the way for the coming of Christ by demonstrating the dire need for His saving work. As Paul has taught in Romans 5, sin existed before the Law [see note Romans 5:14], but men did not recognize it as transgression until the Law came [remembering that transgression is the violation of a known law).

Paul made similar statements regarding the purpose of the Law in Romans 7...

7:7 What shall we say then? Is the Law sin? May it never be! On the contrary, I would not have come to know sin except through the Law; for I would not have known about coveting if the Law had not said, "YOU SHALL NOT COVET."... 7:13 Therefore did that which is good become a cause of death for me? May it never be! Rather it was sin, in order that it might be shown to be sin by effecting my death through that which is good, that through the commandment sin might become utterly sinful. (See notes Romans 7:7;  7:13)

Harrison makes an interesting comment regarding this revelatory purpose of the Law...

This function of the law-viz., to increase transgression-was not recognized in rabbinic Judaism (H.J. Schoeps, Paul [Philadelphia: Westminster Press, 1961], p. 174). From the Sermon on the Mount, however, it appears that Jesus sought to apply the law in just this way, to awaken a sense of sin in those who fancied they were keeping the law tolerably well but had underestimated its searching demands and the sinfulness of their own hearts.  (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

So that (2443) (hina) introduces a purpose clause, and (as just alluded to) in this case introduces the explanation of God's purpose for the Law.

The transgression might increase - He is not saying that God introduced the Law because He wanted to make us sin more, but that He wanted us to be more aware of our sins and the fact that we had a totally sinful nature. (See more detailed discussion of this function of the law below) Notice that Paul does not say transgressions (plural) but transgression (singular), which is a reference to the sin of Adam. How do we arrive at that conclusion? Notice that six times in Romans 5 Paul uses transgression and in each use it is in the singular and in each context, transgression refers to the sin of Adam. Observe the uses and see if you do not agree...

Romans 5:15 But the free gift is not like the transgression. For if by the transgression of the one the many died, much more did the grace of God and the gift by the grace of the one Man, Jesus Christ, abound to the many.

Romans 5:16 And the gift is not like that which came through the one who sinned; for on the one hand the judgment arose from one transgression resulting in condemnation, but on the other hand the free gift arose from many transgressions resulting in justification.

Romans 5:17 For if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ.

Romans 5:18 So then as through one transgression there resulted condemnation to all men, even so through one act of righteousness there resulted justification of life to all men.

To review, every human being has inherited a sin nature from Adam, and the effect of the Law is to stimulate our sin nature so that we commit sins. The commission of personal sins shows that we all have a sin nature. Recall that there are 3 aspects of sin - Inherited Sin (Sin Nature, Sinful Nature, original sin), Imputed Sin and Personal Sins (See discussion and chart on these three aspects of sin)

One might think of the Law as like a mirror one uses to see dirt on one's face. The mirror is not designed to remove the dirt but only to reveal it. Or think of the Law as like a carpenter's plumbline. Plumblines are not meant to straighten the building but to tell one how crooked it is and where the change needs to be made. The Law fulfills its purpose when it makes men realize the full sense of how sinful they are as they see the sins they commit.

Martin Luther wasn't far off when he said that the function of Law was not to justify but to terrify!

Transgression (3900) (paraptoma from para = aside + pipto = fall) is literally a falling aside or beside to stumble on something (so as to loose footing) and in its figurative ethical usage (all uses in the NT) it describes a "false step", a violation of moral standards or a deviation from living according to what has been revealed as the right way to live. Paraptoma is a false step out of the appointed way, a trespass on forbidden ground, a stepping out of line of true conduct, a deviation from truth and uprightness. Paraptoma describes  what a person has done in transgressing the will and law of God by some false step or failure.

Paraptoma is akin to parapipto, to fall beside a person or thing, to fall away, to deviate from the right path, or to turn aside (see note Hebrews 6:6). The basic idea of paraptoma is that of stumbling or falling so as to lose one's footing

The NAS translates paraptoma with 2 words, either as transgression or trespass (derived from Old French - tres =across [Latin - trans] + passer = to pass. Thus trespass means to make inroads upon the property, territory, or rights of another and implies an unwarranted, unlawful, or offensive intrusion).

The Hebrew word (pesha' - 6588) translated as “trespass” means “a stepping aside from the (correct) path” (Ge 31:36; Ex. 22:9), but the Septuagint  does not use paraptoma to translate pesha'.

Thayer writes that paraptoma means

1. properly, a fall beside or near something; but nowhere found in this sense.

2. tropically, a lapse or deviation from truth and uprightness; a sin, misdeed (R. V. trespass, `differing from hamartema in figure not in force'

Vine writes that paraptoma

primarily “a false step, a blunder” (para, “aside,” pipto, “to fall”), then “a lapse from uprightness, a sin, a moral trespass, misdeed,” is translated “fall” (KJV) in Romans 11:11 (note), of the sin and “downfall” of Israel in their refusal to acknowledge God’s claims and His Christ; by reason of this the offer of salvation was made to Gentiles... (Vine, W E: Vine's Complete Expository Dictionary of Old and New Testament Words. 1996. Nelson)

Paraptoma conveys the idea of a false step and so is translated a transgression (transgress in English means to to go beyond or overstep a limit or boundary and is from Latin  trans- across + gradi = to step).

There is a subtle distinction between sin and transgression -- The idea behind transgression is that we have crossed a line, challenging God's boundaries. The idea behind sin  is that we have missed a mark, God's standard that calls for perfection, every time!

NIDNTT says that in Classical Greek...

the noun paraptoma (Polybius onwards) means oversight, error, mistake (unintentional). Here the originally fig. sense was that someone deviated to the one side or the other.  (Brown, Colin, Editor. New International Dictionary of NT Theology. 1986. Zondervan) 

ISBE comments that...

As in Levitical law and Jesus’ teachings, Paul noted that a trespass can have corporate implications. The entire human race experienced vicariously the trespass of Adam (Ro 5:15 note). In like manner, because of the trespass of Israel the message of salvation through Jesus came to the Gentiles (Romans 11:11 note). (Bromiley, G. W. The International Standard Bible Encyclopedia, Revised. Wm. B. Eerdmans)

Vincent has this note on  paraptoma used in Matthew 6:14 writing that...

The Lord here uses another word for sins, and still another (hamartias) appears in Luke’s version of the prayer, though he also says, “every one that is indebted to us.” There is no difficulty in supposing that Christ, contemplating sins in general, should represent them by different terms expressive of different aspects of wrong-doing. This word is derived from parapipto, to fall or throw one’s self beside. Thus it has a sense somewhat akin to hamartia, of going beside a mark, missing. In classical Greek the verb (parapipto) is often used of intentional falling, as of throwing one’s self upon an enemy; and this is the prevailing sense in biblical Greek, indicating reckless and willful sin (see 1 Chr 5:25; 10:13; 2 Chr 26:18; 29:6, 19; Ezek. 14:13; 18:26). It does not, therefore, imply palliation or excuse. It is a conscious violation of right, involving guilt, and occurs therefore, in connection with the mention of forgiveness (see notes Romans 4:25; Romans 5:16; Colossians 2:13; Ephesians 2:1, 2:5). Unlike parabasis (transgression), which contemplates merely the objective violation of law, it carries the thought of sin as affecting the sinner, and hence is found associated with expressions which indicate the consequences and the remedy of sin (see notes Romans 4:25; Romans 5:15; 5:17 Ephesians 2:1)  (Vincent, M. R. Word Studies in the New Testament).

Paraptoma is used 19 times in the NT...

Matthew 6:14 (note) "For if you forgive men for their transgressions, (false steps or faults against others) your heavenly Father will also forgive you.
Matthew 6:1  (note) "But if you do not forgive men, then your Father will not forgive your transgressions. (false steps or faults against God where the repetition in this way brings out the severity of faults against others.)

Mark 11:25 "And whenever you stand praying, forgive, if you have anything against anyone; so that your Father also who is in heaven may forgive you your transgressions. (against whom one transgresses here is not specified).

Romans 4:25  (note) He who was delivered up because of our transgressions, and was raised because of our justification.

Romans 5:15  (note) But the free gift is not like the transgression. For if by the transgression of the one the many died, much more did the grace of God and the gift by the grace of the one Man, Jesus Christ, abound to the many.

Romans 5:16  (note) And the gift is not like that which came through the one who sinned; for on the one hand the judgment arose from one transgression resulting in condemnation, but on the other hand the free gift arose from many transgressions resulting in justification.
Romans 5:17  (note) For if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ.
Romans 5:18  (note) So then as through one transgression there resulted condemnation to all men, even so through one act of righteousness there resulted justification of life to all men.

Romans 5:20  (note) And the Law came in that the transgression (speaking here of the totality of sin) might increase; but where sin increased, grace abounded all the more (Comment: Bauder in NIDNTT writes that "As in the OT, it is used as one of several words for sin, but emphasizes strongly the deliberate act (only in Romans 5:20 is it used of a universal fact) with its fateful consequences. Hence, figuratively it means an action through which man falls and loses the position that God gave him. Thus trespasses committed by one man against another directly affect man’s relation to God and in the final judgment provide the standard by which he is judged (Matt. 6:14 f. par. Lk. 11:25f.). Thus a man must be helped to put any failure right (Gal. 6:1). The first sinful act at the beginning (Rom. 5:15ff.; cf. Wis. 10:1) brought in its train a mass of sin and woe (Rom. 5:18,20), and even death (5:15, 17f.), and that in such a way that even before his physical death man was in the power of death (Eph. 2:1, 5; Col. 2:13). Thus Christ was given up to death (Rom. 4:25) in order that we might receive forgiveness for our sins (2 Cor. 5:19; Eph. 1:7; Col. 2:13). According to Rom. 11:11f., Israel’s fall consists in its rejection of the gospel.) (Ibid) 

Romans 11:11  (note) I say then, they did not stumble so as to fall, did they? May it never be! But by their transgression salvation has come to the Gentiles, to make them jealous.
Romans 11:12  (note) Now if their transgression be riches for the world and their failure be riches for the Gentiles, how much more will their fulfillment be!

2 Corinthians 5:19 namely, that God was in Christ reconciling the world to Himself, not counting their trespasses against them, and He has committed to us the word of reconciliation.

Galatians 6:1 Brethren, even if a man is caught in any trespass, you who are spiritual, restore such a one in a spirit of gentleness; each one looking to yourself, lest you too be tempted.

Ephesians 1:7  (note) In Him we have redemption through His blood, the forgiveness of our trespasses, according to the riches of His grace,

Ephesians 2:1  (note) And you were dead in your trespasses and sins,

Ephesians 2:5  (note) even when we were dead in our transgressions, made us alive together with Christ (by grace you have been saved),

Colossians 2:13  (note) And when you were dead in your transgressions and the uncircumcision of your flesh, He made you alive together with Him, having forgiven us all our transgressions,

There are 14 uses of paraptoma in the Septuagint (LXX) (Job 35:15; 36:9; Ps 19:12; 22:1; Ezek 3:20; 14:11, 13; 15:8; 18:22, 24, 26; 20:27; Dan 4:27; 6:4, 22; Zech 9:5)

Psalm 19:12 Who can discern his errors (Lxx = paraptoma - transgressions? Acquit me of hidden faults.

Spurgeon comments on this passage writing that David's

question is its own answer. It rather requires a note of exclamation than of interrogation. By the law is the knowledge of sin, and in the presence of divine truth, the psalmist marvels at the number and heinousness of his sins. He best knows himself who best knows the Word, but even such an one will be in a maze of wonder as to what he does not know, rather than on the mount of congratulation as to what he does know. We have heard of a comedy of errors, but to a good man this is more like a tragedy. Many books have a few lines of errata at the end, but our errata might well be as large as the volume if we could but have sense enough to see them. Augustine wrote in his older days a series of Retractations; ours might make a library if we had enough grace to be convinced of our mistakes and to confess them. (Spurgeon's note)

Ezekiel 14:13 (note) "Son of man, if a country sins against Me by committing (ma'al - 4603) unfaithfulness (ma'al - 4603), and I stretch out My hand against it, destroy its supply of bread, send famine against it, and cut off from it both man and beast (Comment: In Ezekiel parapipto and paraptoma are used repeatedly and almost always translate the Hebrew verb ma'al which describes the breaking or violation of religious law as a conscious act of treachery, the victim of this breach being God.)

Might increase (4121) (pleonazo from pleion = more) means to cause to increase and suggests an abundance. It means to become more and more so as to be present in abundance.

Guzik has an interesting illustration of the effect of the Law writing that...

The flaws in a precious stone abound when contrasted with a perfect stone, or when put against a contrasting backdrop. God’s perfect law exposes our flaws, and makes our sin abound. There is another way that the law makes sin abound. Because of the sinfulness of my heart, when I see a line drawn I want to cross over it. In this sense, the law makes sin abound because it draws many clear lines between right and wrong that my sinful heart wants to break. Therefore, the law makes me sin more - but not because there is anything wrong in the law, only because there is something deeply wrong in the human condition.