Reformed paedobaptists introduced the concept of substance and accidents into the discussion of covenant theology and wound up creating a rather convoluted mess of things.
The substance/accidents distinction goes back to Aristotle. A simple summary:
Aristotle made a distinction between the essential and accidental properties of a thing. For example, a chair can be made of wood or metal, but this is accidental to its being a chair: that is, it is still a chair regardless of the material from which it is made. To put this in technical terms, an accident is a property which has no necessary connection to the essence of the thing being described.
To take another example, all bachelors are unmarried: this is a necessary or essential property of what it means to be a bachelor. A particular bachelor may have brown hair, but this would be a property particular to that individual, and with respect to his bachelorhood it would be an accidental property.
The concept is liable to abuse. The Roman Catholic Church has used it to explain transubstantiation.
In the first place, the holy Synod teaches, and openly and simply professes, that, in the august sacrament of the holy Eucharist, after the consecration of the bread and wine, our Lord Jesus Christ, true God and man, is truly, really, and substantially contained under the species [accidents] of those sensible things…
And because that Christ, our Redeemer, declared that which He offered under the species of bread to be truly His own body, therefore has it ever been a firm belief in the Church of God, and this holy Synod doth now declare it anew, that, by the consecration of the bread and of the wine, a conversion is made of the whole substance of the bread into the substance of the body of Christ our Lord, and of the whole substance of the wine into the substance of His blood; which conversion is, by the holy Catholic Church, suitably and properly called Transubstantiation…
CANON lI.-If any one saith, that, in the sacred and holy sacrament of the Eucharist, the substance of the bread and wine remains conjointly with the body and blood of our Lord Jesus Christ, and denieth that wonderful and singular conversion of the whole substance of the bread into the Body, and of the whole substance of the wine into the Blood-the species [accidents] Only of the bread and wine remaining-which conversion indeed the Catholic Church most aptly calls Transubstantiation; let him be anathema.
In The Babylonian Captivity of the Church, Luther notes
2.23 The second captivity of this sacrament is less grievous so far as the conscience is concerned, yet the very gravest danger threatens the man who would attack it, to say nothing of condemning it. Here I shall be called a Wycliffite and a heretic a thousand times over. But what of that? Since the Roman bishop has ceased to be a bishop and become a tyrant, I fear none of his decrees, for I know that it is not in his power, nor even in that of a general council, to make new articles of faith. Years ago, when I was delving into scholastic theology, the Cardinal of Cambrai gave me food for thought, in his comments on the fourth Book of the Sentences, where he argues with great acumen that to hold that real bread and real wine, and not their accidents only, are present on the altar, is much more probable and requires fewer unnecessary miracles – if only the Church had not decreed otherwise. When I learned later what church it was that had decreed this – namely, the Church of Thomas, i.e., of Aristotle – I waxed bolder, and after floating in a sea of doubt, at last found rest for my conscience in the above view – namely, that it is real bread and real wine, in which Christ’s real flesh and blood are present, not otherwise and not less really than they assume to be the case under their accidents. I reached this conclusion because I saw that the opinions of the Thomists, though approved by pope and council, remain but opinions and do not become articles of faith, even though an angel from heaven were to decree otherwise. For what is asserted without Scripture or an approved revelation, may be held as an opinion, but need not be believed. But this opinion of Thomas hangs so completely in the air, devoid of Scripture and reason, that he seems here to have forgotten both his philosophy and his logic. For Aristotle writes about subject and accidents so very differently from St. Thomas, that I think this great man is to be pitied, not only for drawing his opinions in matters of faith from Aristotle, but for attempting to base them on him without understanding his meaning – an unfortunate superstructure upon an unfortunate foundation…
2.26 Therefore it is an absurd and unheard-of juggling with words, to understand “bread” to mean “the form, or accidents of bread,” and “wine” to mean “the form, or accidents of wine.” Why do they not also understand all other things to mean their forms, or accidents? Even if this might be done with all other things, it would yet not be right thus to emasculate the words of God and arbitrarily to empty them of their meaning.
Responding to Anabaptists
In an attempt to maintain the Constantinian concept of a state church founded upon infant baptism, reformed theologians stole from the Roman playbook and called upon Aristotle. Arguing that the Old and New Covenants are actually the same covenant, Bullinger says
[T]he nomenclature of the old and new covenant, spirit, and people did not arise from the very essence (substantia) of the covenant but from certain foreign and unessential things (accidentibus) because the diversity of the times recommended that now this, now that be added according to the [difference] of the Jewish people. These additions (accessere) did not exist as perpetual and particularly necessary things for salvation, but they arose as changeable things according to the time, the persons, and the circumstances. The covenant itself could easily continue without them. 
Joshua Moon notes “Bullinger’s reading, and the positing of a unity of substance and contrast of accidents, shows what will emerge as the boundary markers of Reformed thought on the subject. Such language becomes common for the Reformed and will influence the whole of the tradition through the period of orthodoxy and into the contemporary Reformed world.”
Olevianus was a trained humanist as well as a theologian. He learned Aristotle at university and particularly the Organon. As part of his education he learned the traditional Christian appropriation of the distinction between the substance of a thing, i.e., its essence, and its accidents or external appearance.We make this distinction all the time. If you have a smart phone you probably have some sort of cover. The cover is not the phone. It is accidental to the phone. The same is true of your computer. The outer shell that houses your computer isn’t actually the computer. Things like the motherboard, those are the computer… The substance of a thing is what makes it what it is, the thing without which it doesn’t exist. The accidents or circumstances are the administration of the covenant of grace.
Calvin followed the same play.
All this leads to the conclusion, that the difference between us and the ancient fathers lies in accidents, not in substance. In all the leading characters of the Testament or Covenant we agree: the ceremonies and form of government, in which we differ, are mere additions.
Both covenants [are] truly one, though differently administered… The covenant made with all the fathers is so far from differing from ours in reality and substance, that it is altogether one and the same: still the administration differs.
J.V. Fesko notes
What changes, therefore, in the transition from the OT to the NT is not the covenant, but rather the form or administration of the covenant (2.11.13). Here then is what one may describe as Aristotelian language in the use of the distinction between substance and form, which was commonplace in the theology of Calvin’s day.
Cornelius Venema summarizes
When Calvin and subsequent Reformed theologians employ the language of “substance” and “form” or “accidents” to refer to the distinct administrations of the one covenant of grace throughout history, they are employing a traditional category distinction from the philosophy of Aristotle. “Substance” refers to “what makes something what it is,” “accidents” refers to what belongs “contingently” to something.
Administration = Accidents
The accidents of the covenant of grace were identified with its “administration,” referring to various ordinances and ceremonies. WCF 7.5-6 identify these as “promises, prophecies, sacrifices, circumcision, the paschal lamb, and other types and ordinances delivered to the people of the Jews, all foresignifying Christ to come” as well as “the preaching of the Word, and… the sacraments of Baptism and the Lord’s Supper.”
Substance = Salvation in Christ
The substance, then, refers to what is being administered: salvation in Christ.
[T]he comparison made by the Apostle refers to the form rather than to the substance; for though God promised to them the same salvation which he at this day promises to us, yet neither the manner nor the character of the revelation is the same or equal to what we enjoy.
-Calvin, Commentary Hebrews 8:6
How did the Olevianus and others define the substance or essence of the covenant of grace? “I will put my law in your midst, and I will write my law in your heart and I will be your God and you will be my people.” Embedded in this prophetic articulation of the covenant of grace is essentially or substantially the same promise he had made to Adam, after the fall (Gen 3), to Noah (Gen 6), and to Abraham (Gen 17). Embedded in that re-articulation is the ancient promise to send a redeemer who would turn away the wrath we earned and to earn righteousness for all his people. This, Olevianus would go on to say is the first benefit of the covenant of grace: “free forgiveness of sins in Christ,” i.e., unconditional acceptance with God by grace alone, through faith alone, in Christ alone…
When our theologians, whether Olevianus in the 16th century or Witsius in the 17th century, wrote about the “substance of the covenant” they were writing about the same way God has always saved and sanctified his people whether under Adam, Noah, Abraham, Moses, David or Christ. There is a unified covenant of grace.
-R. Scott Clark, What Is The Substance Of The Covenant Of Grace? (2)
Accidents = Shadows?
I’m not certain when it was first articulated, but an important twist occurs as the concept is further developed.
WCF 7.6 Under the Gospel, when Christ, the substance, was exhibited, the ordinances in which this covenant is dispensed are the preaching of the Word, and the administration of the sacraments of Baptism and the Lord’s Supper: which, though fewer in number, and administered with more simplicity, and less outward glory, yet, in them, it is held forth in more fullness, evidence, and spiritual efficacy, to all nations, both Jews and Gentiles; and is called the New Testament. There are not therefore two covenants of grace, differing in substance, but one and the same, under various dispensations.
Notice the two uses of the word “substance”. The argument is that because Christ is the substance, there are not two different covenants, but only one. However, note Scripture reference : “Therefore let no one pass judgment on you in questions of food and drink, or with regard to a festival or a new moon or a Sabbath. These are a shadow of the things to come, but the substance belongs to Christ.” (Colossians 2:16-17 ESV) Thus the first mention of “substance” refers to a shadow/substance distinction, or what we could call a type/anti-type distinction. One thing points forward to something else. However, the second use of “substance” refers to the substance/accidents distinction.
Are these two distinctions the same thing? Was Paul using Aristotelian categories when he spoke of shadows and substance? No. They are two different concepts. Two different distinctions. The shadow/substance distinction refers to a way of teaching or speaking about something by way of analogy or comparison. The substance/accidents distinction refers to defining the essence of something.
The Westminster tradition has conflated these two things and built a labyrinth around themselves that they are now trapped in. It has stunted their typology. The recent OPC Report on Republication addresses this point (in order to show how Kline’s typology is contrary to the WCF).
According to our doctrinal standards the substance of the covenant of grace is Christ. The covenant was fulfilled “under the gospel, when Christ, the substance, was exhibited” (WCF 7.6). Christ supplies the substance (or blessings) of the covenant of grace due to the dignity of his person and the merit of his work… Whether we are speaking of the types and pictures of Christ in the old covenant or the reality and fullness of Christ in the new, what is applied to God’s elect, in principle, is the same. Although the ceremonies, sacrifices, and ordinances of the Mosaic covenant were types of Christ, the efficacy of what they pictured was communicated through them to the elect of Israel…
However, it is also true that some Reformed theologians have seen the idea of substance in a more technical way; namely, the core condition that governs the covenant. Thus, when the condition is essentially the same, the covenant is also essentially the same; and when the condition differs, so does the essence of the covenant. For example, Zacharias Ursinus argues that the “substance of the covenant” is “the principal conditions” of the covenant… The confession seems to communicate this basic idea when it states that the Old and New Testament are not “two covenants of grace, differing in substance, but one and the same under various dispensations.”…
The confession addresses these differences [between the Old and New Covenants] by the way in which the covenant itself is administered, and by the way in which the blessings of the covenant are enjoyed. It does this by organizing these two issues through its unified treatment and emphasis on typology… The covenant of grace was administered in the time of the law “by promises, prophecies, sacrifices, circumcision, the paschal lamb, and other types and ordinances.” The phrase, “other types and ordinances” shows that typology functions as a general rubric to summarize the symbols and ordinances of the old covenant. The standards remind us that those types were “sufficient and efficacious” for the time of the law and by them believing Israelites enjoyed the “full remission of sins, and eternal salvation” (WCF 7.5). Yet this is true only because they were more than symbols for that covenant administration. They also functioned as types of the fullness to be unveiled with Christ’s coming. Their ultimate efficacy is dependent upon their functioning as types.
By adding obedience to the ceremonial law to the essential condition of the covenant, the subservient covenant position gives Mosaic typology a fundamentally works-based character, rather than an evangelical one. Proponents did not deny that these various types also signified spiritual benefits, but they insisted that they only did so “secondarily” or indirectly, while their primary reference was to temporal things promised in the covenant.169
 Cameron put it this way: “The Sacraments, Sacrifices, and Ceremonies of the Old Testament did set forth Christ, and the Benefits by Christ; not primarily, but secondarily…but the Sacraments of the New Covenant do shew forth Christ primarily, and that clearly” (as translated by Samuel Bolton in his True Boundes, 399). Thus circumcision primarily signified the separation between the seed of Abraham and the rest of the nations and sealed to them the earthly promise. The Passover primarily signified the passing over of the destroying Angel. The sacrifices and washings primarily represented only a carnal holiness. Only secondarily did these benefits signify Christ.
From a confessional viewpoint, the basic weakness here is that it reverses the true biblical priority of Christ as the substance and primary signification of these types and shadows. According to our standards, the purpose of these various types and ordinances was to function as an aspect of the covenant of grace, being means of administering the eternal and salvific blessings procured by Christ (WCF 7.5, 8.6, 17.5). He is the “substance” of the types and ordinances (not merely their secondary referent), even as he is the substance of God’s covenant of grace (WCF 7.6), while all else remains secondary or accidental. The subservient covenant effectively reverses this in insisting that these types primarily signify temporal benefits, and only secondarily signify Christ. As John Cameron stated, the subservient covenant leads to Christ only “indirectly” whereas the covenant of grace leads to him directly. It is difficult to harmonize the idea that Christ was the “substance” of all these types and ordinances and at the same time only their secondary referent…
[A]nything that functions as an “administration” of the covenant of grace must, in fact, administer grace to those who are under it. Such it is with the other types, ceremonies, and other ordinances delivered to the Jews. The administrative aspects of the old covenant were to function as the “outward and ordinary means whereby Christ communicates” to Israel “the benefits of redemption” (SC 88)… [T]ypology is a subset of the broader category of the administration of the covenant… According to our standards, typology is an aspect of the administration of the covenant of grace in the Old Testament, which in turn is described as the outward means of the Old Testament era for communicating grace to the elect of that era. Saving grace was not simply administered merely as a consequence or by-product of these types.275 Rather, saving grace was present by and in these types, and in this way communicated grace to believers.276 In terms of our confessional definitions, to say that something is an administration of grace means that grace is communicated by and in that thing.
 This seems to be the distinctive typological construction of the subservient covenant position, discussed above.
For more on this view, listen to an interview on the Reformed Forum with Lane Tipton, one of the co-authors of the Report.
Subservient Covenant Typology
I once had a Presbyterian say to me
I always thought Owen’s claims about the Poverty of Types meant he was on a pretty different page from the rest of the presbyterians
“Such was the poverty of the types that no one of them could so much as shadow out or represent all that advantage which we really enjoy and therefore they were multiplied and the work distributed amongst them which they were to represent. This made them a yoke, and that grievous and burdensome. The way of teaching in them and by them was hard and obscure, as well as their observation was difficult. It was a hard thing for them to learn the love, grace, and mind of God by them. God revealed himself in them πολυμερῶς, by many parts and pieces, according as they were capable to receive impression from and make representation of divine wisdom, goodness, and grace; whence our apostle says, that the law had but σκίαν, “a shadow,” and not αὐτὴν τὴν εἰκόνα πραγμάτων, Hebrews 10:1, — “the image itself of things.” It had some scattered shades, which the great limner had laid the foundation of symmetry in, but so as to be discernible only unto his own infinite wisdom. A perfect image, wherein all the parts should exactly answer unto one another, and so plainly represent the thing intended, that it had not. Now, it was a work beyond their wisdom, out of these scattered pieces and parts of revelation, especially being implanted on carnal things, to gather up the whole of the grace and good-will of God. But in Christ Jesus God hath gathered all into one bead, Ephesians 1:10, wherein both his person and grace are fully and at once represented.”
Owen, like 1689 Federalism, held to a version, or refinement of the subservient covenant view, which recognizes that the Old and New Covenants are two different, distinct covenants – not the same covenant. Thus the Old and New are not one in essence or substance. However, it is important to understand that Owen and 1689 Federalism do affirm that Christ is the substance of the the types and shadows of the Old Covenant and that men in the Old Testament were saved through belief in the gospel revealed by those types. They simply recognize that those two uses of “substance” are two different concepts.
Having noted these things, we may consider that the Scripture does plainly and expressly make mention of two testaments, or covenants, and distinguish between them in such a way as can hardly be accommodated by a twofold administration of the same covenant…Wherefore we must grant two distinct covenants, rather than merely a twofold administration of the same covenant, to be intended. We must do so, provided always that the way of reconciliation and salvation was the same under both. But it will be said, and with great pretence of reason, for it is the sole foundation of all who allow only a twofold administration of the same covenant, ’That this being the principal end of a divine covenant, if the way of reconciliation and salvation is the same under both, then indeed they are the same for the substance of them is but one.’ And I grant that this would inevitably follow, if it were so equally by virtue of them both. If reconciliation and salvation by Christ were to be obtained not only under the old covenant, but by virtue of it, then it must be the same for substance with the new. But this is not so; for no reconciliation with God nor salvation could be obtained by virtue of the old covenant, or the administration of it, as our apostle disputes at large, though all believers were reconciled, justified, and saved, by virtue of the promise, while they were under the old covenant.
Having shown in what sense the covenant of grace is called “the new covenant,” in this distinction and opposition to the old covenant, so I shall propose several things which relate to the nature of the first covenant, which manifest it to have been a distinct covenant, and not a mere administration of the covenant of grace.
That which before lay hid in promises, in many things obscure, the principal mysteries of it being a secret hid in God himself, was now brought to light; and that [new] covenant which had invisibly, in the way of a promise, put forth its efficacy under types and shadows, was now solemnly sealed, ratified, and confirmed, in the death and resurrection of Christ
-Owen on Hebrews 8:6
The types and shadows of the Old Covenant revealed the gospel and people were saved by believing that gospel, but the Old Covenant did not therefore save them because it did not establish union with Christ. The New Covenant is our union with Christ. The Old Covenant types were the means that God used to reveal the gospel but it was the New Covenant union established in the effectual call that saved the elect living under the Old Covenant.
[Side Note: Klineans do not properly understand the use of Aristotelian “substance” in WCF 7.6 and the reformed tradition to affirm that the Old and the New are the same covenant. They reject that idea and say the Old and New are two distinct covenants, but they still try to argue that they affirm 7.6. They simply don’t understand what 7.6 is saying – and part of that is because 7.6 conflates two different ideas about substance: one they affirm and one they do not. For more on this point, see Kline on “Administration of the Covenant of Grace” and Episodes 4-6 of the Glory Cloud Podcast. Owen properly understood the meaning of terms and therefore rejected WCF 7.6.]
The OPC Report notes “It is difficult to harmonize the idea that Christ was the ‘substance’ of all these types and ordinances and at the same time only their secondary referent.” This difficulty is only created by the mistaken application of substance/accidents. Presbyterians who stumble at this point would do well to listen to Augustine, who addresses what he sees as an error on their part.
City of God
Book XVII: The history of the city of God from the kings and prophets to Christ.
Chapter 3.—Of the Three-Fold Meaning of the Prophecies, Which are to Be Referred Now to the Earthly, Now to the Heavenly Jerusalem, and Now Again to Both.
Wherefore just as that divine oracle to Abraham, Isaac, and Jacob, and all the other prophetic signs or sayings which are given in the earlier sacred writings, so also the other prophecies from this time of the kings pertain partly to the nation of Abraham’s flesh, and partly to that seed of his in which all nations are blessed as fellow-heirs of Christ by the New Testament, to the possessing of eternal life and the kingdom of the heavens. Therefore they pertain partly to the bond maid who gendereth to bondage, that is, the earthly Jerusalem, which is in bondage with her children; but partly to the free city of God, that is, the true Jerusalem eternal in the heavens, whose children are all those that live according to God in the earth: but there are some things among them which are understood to pertain to both,—to the bond maid properly, to the free woman figuratively. (Gal 4:22-31)
Therefore prophetic utterances of three kinds are to be found; forasmuch as there are some relating to the earthly Jerusalem, some to the heavenly, and some to both. I think it proper to prove what I say by examples. The prophet Nathan was sent to convict king David of heinous sin, and predict to him what future evils should be consequent on it. Who can question that this and the like pertain to the terrestrial city, whether publicly, that is, for the safety or help of the people, or privately, when there are given forth for each one’s private good divine utterances whereby something of the future may be known for the use of temporal life? But where we read, “Behold, the days come, saith the Lord, that I will make for the house of Israel, and for the house of Judah, a new testament: not according to the testament that I settled for their fathers in the day when I laid hold of their hand to lead them out of the land of Egypt; because they continued not in my testament, and I regarded them not, saith the Lord. For this is the testament that I will make for the house of Israel: after those days, saith the Lord, I will give my laws in their mind, and will write them upon their hearts, and I will see to them; and I will be to them a God, and they shall be to me a people;” (Heb 8:8-10) without doubt this is prophesied to the Jerusalem above, whose reward is God Himself, and whose chief and entire good it is to have Him, and to be His. But this pertains to both, that the city of God is called Jerusalem, and that it is prophesied the house of God shall be in it; and this prophecy seems to be fulfilled when king Solomon builds that most noble temple. For these things both happened in the earthly Jerusalem, as history shows, and were types of the heavenly Jerusalem. And this kind of prophecy, as it were compacted and commingled of both the others in the ancient canonical books, containing historical narratives, is of very great significance, and has exercised and exercises greatly the wits of those who search holy writ. For example, what we read of historically as predicted and fulfilled in the seed of Abraham according to the flesh, we must also inquire the allegorical meaning of, as it is to be fulfilled in the seed of Abraham according to faith. And so much is this the case, that some have thought there is nothing in these books either foretold and effected, or effected although not foretold, that does not insinuate something else which is to be referred by figurative signification to the city of God on high, and to her children who are pilgrims in this life. But if this be so, then the utterances of the prophets, or rather the whole of those Scriptures that are reckoned under the title of the Old Testament, will be not of three, but of two different kinds. For there will be nothing there which pertains to the terrestrial Jerusalem only, if whatever is there said and fulfilled of or concerning her signifies something which also refers by allegorical prefiguration to the celestial Jerusalem; but there will be only two kinds one that pertains to the free Jerusalem, the other to both. But just as, I think, they err greatly who are of opinion that none of the records of affairs in that kind of writings mean anything more than that they so happened, so I think those very daring who contend that the whole gist of their contents lies in allegorical significations. Therefore I have said they are threefold, not two-fold. Yet, in holding this opinion, I do not blame those who may be able to draw out of everything there a spiritual meaning, only saving, first of all, the historical truth. For the rest, what believer can doubt that those things are spoken vainly which are such that, whether said to have been done or to be yet to come, they do not beseem either human or divine affairs? Who would not recall these to spiritual understanding if he could, or confess that they should be recalled by him who is able?”
A Work on the Proceedings of Pelagius.
Chapter 14.—Examination of This Point. The Phrase “Old Testament” Used in Two Senses. The Heir of the Old Testament. In the Old Testament There Were Heirs of the New Testament.
…In that testament, however, which is properly called the Old, and was given on Mount Sinai, only earthly happiness is expressly promised. Accordingly that land, into which the nation, after being led through the wilderness, was conducted, is called the land of promise, wherein peace and royal power, and the gaining of victories over enemies, and an abundance of children and of fruits of the ground, and gifts of a similar kind are the promises of the Old Testament. And these, indeed, are figures of the spiritual blessings which appertain to the New Testament; but yet the man who lives under God’s law with those earthly blessings for his sanction, is precisely the heir of the Old Testament, for just such rewards are promised and given to him, according to the terms of the Old Testament, as are the objects of his desire according to the condition of the old man. But whatever blessings are there figuratively set forth as appertaining to the New Testament require the new man to give them effect. And no doubt the great apostle understood perfectly well what he was saying, when he described the two testaments as capable of the allegorical distinction of the bond-woman and the free,—attributing the children of the flesh to the Old, and to the New the children of the promise: “They,” says he, “which are the children of the flesh, are not the children of God; but the children of the promise are counted for the seed.” (Rom 9:8) The children of the flesh, then, belong to the earthly Jerusalem, which is in bondage with her children; whereas the children of the promise belong to the Jerusalem above, the free, the mother of us all, eternal in the heavens. (Gal 4:25, 26) Whence we can easily see who they are that appertain to the earthly, and who to the heavenly kingdom. But then the happy persons, who even in that early age were by the grace of God taught to understand the distinction now set forth, were thereby made the children of promise, and were accounted in the secret purpose of God as heirs of the New Testament; although they continued with perfect fitness to administer the Old Testament to the ancient people of God, because it was divinely appropriated to that people in God’s distribution of the times and seasons.
The Substance of the Old Covenant
If Christ is not the substance, or essence, of the Old Covenant, then what is? Well, substance refers to the essence of something. So, what is essential to any particular covenant – divine or human? Simply put, the parties and the terms.
The OPC Report quotes Thomas Blake explaining that “a covenant entered by the same parties, upon the same terms and propositions on either hand, is the same covenant.” Thus the “substance” of each biblical covenant could be identified as follows:
- Adamic: between God and Adam, representing all humanity, offering eternal life upon the condition of perfect obedience
- Noahic: between God and Noah, representing all humanity, promising never to flood the earth again without condition (or alternatively upon condition of Noah building and entering the ark)
- Abrahamic: between God and Abraham, representing his carnal offspring, promising to give him numerous physical offspring and the land of Canaan for them to dwell in, and also promising that the Messiah will be born from him and will bless all nations
- Mosaic: between God and Israel, mediated by Moses, promising to bless them in the land of Canaan or to curse them in exile upon condition of their obedience to the law of Moses
- Davidic: between God and David, representing his offspring, promising to make them king of Israel and to bless Israel upon condition of their obedience to the law of Moses
- Redemption: between the Father and the Son, promising to grant the Son an kingdom and a redeemed people upon condition of his active and passive obedience
- New: between God and Christ, representing the elect, promising to pour out his Spirit upon them, granting them faith, justification, sanctification, glorification – all the benefits of union with Christ without any antecedent condition on their part
But I prefer to avoid speaking in terms of “substance” and to just speak about the parties and terms of each covenant. And, when a paedobaptist brother echoes Calvin, saying
[T]he comparison made by the Apostle [in Hebrews 8] refers to the form rather than to the substance… The ceremonies of the law… were merely accidents of the covenant, or at least additions and appendages, and, as they are commonly called, accessories, yet because they were the means of administering it, the name of covenant is applied to them…
I simply echo Luther
Therefore it is an absurd and unheard-of juggling with words, to understand “covenant” to mean “the form, or accidents of the covenant,” and “wine” to mean “the form, or accidents of wine.” Why do they not also understand all other things to mean their forms, or accidents? Even if this might be done with all other things, it would yet not be right thus to emasculate the words of God and arbitrarily to empty them of their meaning.