Typically, when we think of the OC (Old Covenant) we think of only the Mosaic Covenant made with the people of Israel on Mount Sinai, with all of the old laws that were specific to Israel and their culture, the Ten Commandments, and the requirement for sacrifices and circumcision. These ingredients are often thought to be what make up the OC because Jeremiah (31:31-34) refers to the covenant made with Israel upon the exodus out of Egypt with Moses as the mediator (vs. 32). However, to many of our seventeenth century particular Baptist forefathers, the OC included the entire Old Testament as well (from the fall to Christ). They did not view it as a single isolated covenant but as cumulative.
What is the reasoning behind this view of the OC? Pascal Denault puts it this way: “several times, the New Testament presents the covenant between God and Israel (the Old Covenant) rooted both in the covenant with the patriarchs and in the Mosaic Covenant.”  Both Jesus and Paul spoke about circumcision and the Law of Moses as inseparable (Jn. 7:22-23; Gal. 5:3). Other apostles describe circumcision with the heavy weight of the Law of Moses (Ac. 15:5, 10-11). In Acts 7, Stephen starts his overview of the OC with Abraham (covenant of circumcision in verse 8) then leads into preaching about the Mosaic and Davidic Covenants. “But it is clear to me that in substance, the same covenant of ceremonial obedience which was given to Moses when the people came out of the Egypt, the same was given to Adam’s generation.” 
Throughout the first covenant (the OC), Christ was the mediator for His elect (also referred to as the “remnant”), having paid for their sins effectually even before the NC (New Covenant) had been established in the future (Heb. 9:15). Christ came to establish the NC, to fulfill the whole law and then replace the OC in His life as a man, His atoning death on the cross for sin once for all time, and His resurrection from the dead. Therefore, the OC is the whole Old Testament from the time of the fall to Christ. “We begin the economy of the Old Testament immediately upon the fall, and the first promise of grace, and end it in Christ.” 
How was the Old Covenant displayed in the Old Testament?
Before the Fall, “God gave Adam a law of universal obedience written in his heart, and a particular precept of not eating the fruit of the tree of knowledge of good and evil; by which he bound him and all his posterity to personal, entire, exact, and perpetual obedience; promised life upon the fulfilling, and threatened death upon the breach of it, and endued him with power and ability to keep it,” (1689 LBCF 19:1). In concept, God made a covenant of works with Adam unto life.
When tempted, Adam fell into sin, thus, “sin entered into the world, and death through sin, and so death spread to all men, because all sinned,” (Rom. 5:12); Adam broke the covenant. After the fall, God promises redemption through the seed of the woman (Gen. 3:15), but does not formally establish this new covenant to replace the one Adam broke. Therefore, Adam was still under law that was given to him (Gen. 2:16-17).
“The Covenant of Works was primarily made with the First Adam, and all mankind in him…True, there was another Edition or Administration of it given to Israel, which tho’ it was a Covenant of Works, i.e. Do this and live, yet it was not given by the Lord to the same End and Design, as the Covenant was given to our First Parents, viz. It was not given to justifie them, or to give eternal life.” 
Moreover, we see aspects of the OC during the time before the Mosaic Law in the animal sacrifices, “which were at the basis of the Old Covenant (Heb. 7:11), started well before the Levitical priesthood. Sacrifices are present as of Genesis (Gn. 3:21; 4:4; 8:20; 22:13; 46:1) and the Levitical priesthood represents their continuity.”  Therefore, we clearly see language in the beginning of the Torah of a covenant of works and obedience made with Adam before the fall that continued after the fall as well. This covenant was not salvific unto eternal life but was commonly gracious to all who benefited from its blessings.
Abrahamic & Mosaic Covenant
The OC was seen in Abraham on the condition of circumcision along with blessings (land and protection; Gen. 17:8) and curses (you will be cut off from your people; Gen. 17:14). Then after the great exodus out of Egypt, the earthly seed of Abraham (Hebrews) were given a “new” covenant that built off of the one given to Abraham who served as the nation’s head, and did not remove or replace it. Not much was new, however, because the blessings and curses were the same (death and being cut off from your people), man was still under the curse of the law (Ex. 21:1; Gal. 3:10), and the Hebrews were still only made outwardly clean by proper obedience to the law and still considered to be a part of the people as circumcision allowed.
What was new was that it pleased God to give them His eternal moral law that He elaborated and articulated (Ex. 20:1-21; Duet. 10:4; Rom. 2:14, 15), the ceremonial laws pertaining to worship and particular ordinances that foreshadowed as a type of Christ in His gracious acts, pains, and benefits (Lev. 16; Heb. 10:1; Col. 2:17), and judicial laws that were temporal and specific for the nation of Israel for general benefit (1 Cor. 9:8-10). Thus, summed up in the Law of Moses given by God to reveal the character of God and how He ought to be worshipped properly. This law, like the covenant of circumcision, was conditional upon the obedience of man resulting in blessings or curses (Deut. 28). The Law commanded obedience from the heart but offered no way to be able to overcome sin and truly obey God in heart and soul (Deut. 5:29; 6:6; 29:4).
Nehemiah Coxe said, “…the Lord was pleased to draw the first lines of that form of covenant relationship in which the natural seed of Abraham was fully stated (to instate or establish a covenant) by the law of Moses, which was a covenant of works with its condition or terms, ‘Do this and live.’ For although the covenant of grace made with Abraham has in all respects (in point of time and excellency) the precedence to the covenant made with his carnal seed in Isaac’s line, yet in the counsel of God things were so ordered that the full revelation of the covenant of grace, the actual accomplishment of its great promises, and its being filled up with ordinances proper to it, should succeed the covenant made with Israel after the flesh, and replace it on its dissolution when it waxed old and vanished away. Therefore the covenant interest of the natural seed was to be perfected by the law of Moses before the gospel preached to Abraham was unveiled.” 
This does not define the OC as only the Covenant of Circumcision, the covenant with Abraham, but the covenant made with Abraham was the beginning of God’s giving of the law to His people; God gave Abraham and his natural seed positive law, meaning law that does not come naturally to men like the moral law, which was to be circumcised in order to enter into the covenant. Then, when the time came, God provided Israel with another covenant on Sinai which built on the Covenant of Circumcision. The Mosaic Covenant is substantially the same as the preceding Abrahamic Covenant because to be allowed in the camp one must obey God’s commands by being circumcised and made clean according to the law. In order to be in the covenant, one must obey, if one disobeys one is cut off from one’s people and/or dies.
Purpose of the Abrahamic & Mosaic Covenants
The Abrahamic and Mosiac covenants contained a two-fold purpose. One earthly, to have a temporal relationship with God based on works. This relationship was not personal or intimate, nor did it provide salvation for the soul, it was earthly and temporal. If the people obeyed, they were kept safe and God provided for them. If they disobeyed, they were cursed.
The second purpose was spiritual, they served as types and shadows of the NC (Heb. 8:5). Abraham was promised a Covenant of Grace but it was not formally established in him in Genesis 12. “Again, its called the promise, and not the Covenant; and we know that every promise is not a covenant: there being large difference between a promise and a covenant. And now let it be well considered what is here meant by the promise, and that is Gods sending of the Messias, or the seed in whom the Nations should be blessed; and so the sending of a Saviour or Redeemer unto Israel.” 
Likewise, the Seventeenth Century Particular Baptists did not believe the Covenant of Grace was actually established in Genesis 12, it was promised, however, a covenant of works was being formally established. “They saw only one formal Abrahamic Covenant: the Covenant of circumcision established in Genesis 17, all the while clearly differentiating this covenant from the promise (the Covenant of Grace) that God had previously made.” 
Basically, when God said, “…in you all the families of the earth will be blessed” (Gen. 12:3), God promises effectual saving grace to all nations. When God said, “This is My covenant, which you shall keep, between Me and you and your descendants after you: every male among you shall be circumcised” (Gen. 17:10), God establishes a covenant that is based on the obedience of the earthly seed of Abraham. The first is unconditional, there were no conditions given to Abraham for him to obey for that promise to come true, and the second was conditional upon man’s works of obedience according to law. It is in the second that we find within the OC.
Moreover, when Christ established the NC, it is clear, according to Hebrews 8:13, that the OC is now “obsolete.” Jesus includes circumcision within the Law of Moses but not originating in Moses in John 7:22-23. Thus, the Covenant of Circumcision (Acts 7:22) and the Mosaic Covenant, and all of their works-based temporal protection, are obsolete and replaced with the new and better covenant (the NC).
Greg Nichols says, “The old and new covenants directly relate to these visible communities of God’s people. The old covenant promises relate to the organized society of Abraham’s physical descendants circumcised in body. The new covenant promises relate to the organized society of Abraham’s spiritual children circumcised in heart.” 
The overall purpose of the OC was not to make man perfectly righteous before God because no man, due to sin, can be declared righteous before God upon any works or obedience of the man since man cannot obey the whole law perfectly. The purpose of the OC was to point man to the true and only way to be made right with God, and that is Christ. The OC was created to point to the NC in Christ alone to the glory of God alone.
 Pascal Denault, The Distinctiveness Of Baptist Covenant Theology, p. 100.
 Thomas Patient, The Doctrine of Baptism, And the Distinction of the Covenants, beginning of Chapter 10.
 Herman Witsius, The Economy of the Covenants, vol. 1, p. 308.
 Benjamin Keach, The Everlasting Covenant, p. 7.
 Denault, p. 100.
 Nehemiah Coxe, Covenant Theology From Adam to Christ, p. 91.
 John Spilsbury, A Treatise Concerning the Lawfull Subject of Baptisme, p. 26.
 Denault, p. 122.
 Greg Nichols, Covenant Theology, p. 273.