Chapter 5: The Seal and the Mark

Official seals contain three elements that establish authority: a name, a title, and a territory. The Presidential Seal identifies the president’s name, "President of the United States" (title), and the territory over which that authority extends. In ancient times, kings sealed official documents with signet rings pressed into wax, leaving an identifying mark that could not be forged. Without all three elements, a seal lacks legitimacy.

God’s seal follows the same pattern. Just as earthly kings use seals to authenticate decrees and establish their authority, the Creator placed His seal in His law. And it’s found in only one place: the only commandment that tells you who He is, what He does, and where His dominion extends. (Explore the seal’s three elements with the Seal Calculator.)

Seal vs. mark comparison chart: https://theremnantthread.com/studies/seal-vs-mark

Mark enforcement progression (four phases): https://theremnantthread.com/studies/mark-progression

A necessary clarification before proceeding: Sunday-keeping today is not the mark of the beast. The mark becomes a test only when Sunday worship is legally enforced, and refusing the true Sabbath costs you economically. Until that enforcement comes, the test hasn’t come. Sincere Christians in every tradition who keep Sunday without understanding this issue are not condemned. God doesn’t hold people accountable for light they haven’t received (Acts 17:30). This chapter examines what Scripture teaches and what history documents, not to condemn the past but to illuminate the future.

God’s Mark: The Seal in the Sabbath

The Fourth Commandment contains all three elements of God’s official seal:

"Remember the sabbath day, to keep it holy. Six days shalt thou labour, and do all thy work: But the seventh day is the sabbath of the LORD thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: For in six days the LORD made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it."

Exodus 20:8–11

Element 1: Name, "the LORD thy God" (YHWH)

Element 2: Title, "made heaven and earth, the sea, and all that in them is" (Creator/Maker)

Element 3: Territory, "heaven and earth, the sea, and all that in them is" (the entire universe)

Name, title, and territory: the three elements required for any official seal appear together in only one commandment.

The other nine commandments either refer to "the LORD thy God" without identifying which Lord (the first through third) or contain pure moral law with no mention of God at all (the fifth through tenth). Every god claims to be lord. Every religion has commandments. Only the Fourth Commandment identifies the Lawgiver by stating what He did: "For in six days the LORD made heaven and earth, the sea, and all that in them is."

Only the Creator of heaven and earth has authority to command all heaven and earth. Only the Maker of all things has the right to demand obedience from His creation. The Fourth Commandment declares who this God is by stating what He did.

The Sabbath commandment is God’s signature on His law.

The Seal Distinguishes the True God from Counterfeits

Every god claims authority. Every religion has commandments. But only the God of the Bible claims to have made heaven and earth in six days, rested the seventh, and commanded that day as a memorial of His creative power. Only the true Creator can point to creation and say, "This is My seal. This identifies you as Mine."

This is why God calls the Sabbath His sign, His identifying mark:

"Verily my sabbaths ye shall keep: for it is a sign between me and you throughout your generations; that ye may know that I am the LORD that doth sanctify you."

Exodus 31:13

"And hallow my sabbaths; and they shall be a sign between me and you, that ye may know that I am the LORD your God."

Ezekiel 20:20

Scripture calls the Sabbath a "sign." Paul answers whether a sign can function as a seal directly:

"And he received the sign of circumcision, a seal of the righteousness of the faith which he had yet being uncircumcised."

Romans 4:11

Circumcision was both a "sign" (Greek: ÏƒÎ·ÎŒÎ”áż–ÎżÎœ, semeion) and a "seal" (Greek: ÏƒÏ†ÏÎ±ÎłÎŻÏ‚, sphragis). The two terms are not mutually exclusive. A sign that identifies covenant membership functions as a seal of that relationship.1 The Greek word sphragis (seal) in Romans 4:11 is the same word used in Revelation 7:2 for "the seal of the living God." The standard Greek lexicon defines sphragis as "that which confirms or authenticates, attestation, confirmation." Walter Bauer et al., A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000), 980.

If circumcision, the sign of the Abrahamic covenant, was also a seal, then the Sabbath, the sign of the Creation covenant, can function the same way. The Sabbath identifies you as belonging to the Creator. It marks you as one who acknowledges His authority as Maker of heaven and earth.

The seventh-day Sabbath is God’s sign distinguishing those who worship Him from those who worship counterfeits.

When you keep the Sabbath, you wear God’s seal.

The Greek terms themselves reveal the contrast. The word for "seal" in Revelation 7:2 is ÏƒÏ†ÏÎ±ÎłÎŻÎ¶Ï‰ (sphragizo), derived from a root meaning to stake a claim of ownership through a voluntary covenant. The word for "mark" in Revelation 13:16 is Ï‡ÎŹÏÎ±ÎłÎŒÎ± (charagma), meaning an etching or brand placed on property or slaves without their consent. One is accepted freely. The other is imposed by force. The seal identifies willing covenant-keepers. The mark brands conscripted subjects.

The Seal as Protection

The seal is not merely identification. It is protection. Scripture shows God marking His people before destruction falls.

Ezekiel 9 records God’s instruction before Jerusalem’s judgment:

"Go through the midst of the city, through the midst of Jerusalem, and set a mark upon the foreheads of the men that sigh and that cry for all the abominations that be done in the midst thereof."

Ezekiel 9:4

Those with the mark were spared. Those without it were destroyed. Revelation echoes the same pattern: "Hurt not the earth, neither the sea, nor the trees, till we have sealed the servants of our God in their foreheads" (Revelation 7:3). Sealing precedes judgment. Protection comes before destruction.

If the seal identifies those who belong to God, the mark identifies those who belong to another authority. The final choice is not merely "loyal versus disloyal." It is "protected versus unprotected when the wrath is poured out."

The New Testament Connection

Some Christians, when first encountering this teaching about God’s seal in the Sabbath, raise a common objection worth addressing:

Some object: "The seal in the New Testament is the Holy Ghost, not the Sabbath." They cite Ephesians 1:13 ("sealed with that holy Ghost of promise") and Ephesians 4:30 ("whereby ye are sealed unto the day of redemption").

True: the Spirit seals. But the Spirit seals us with something specific.

"For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts."

Hebrews 8:10

The Spirit writes God’s law on hearts. That’s what the New Covenant is: the same law, written on hearts instead of stone. The Spirit doesn’t replace the commandments; the Spirit internalizes them.

The Spirit writes the law of God, including the Fourth Commandment: the one that identifies the Creator and contains His seal.

The Spirit seals us by placing God’s law in our minds and hearts. The Sabbath commandment is part of that law. The two aren’t contradictory; they’re complementary. The Spirit seals through obedience. The Sabbath is how that seal is expressed.

When Revelation 7:3 describes God sealing His servants "in their foreheads," it’s describing what the Spirit does: placing God’s law in the mind. And the commandment that specifically identifies God, revealing His name, title, and dominion, is the fourth.

The Simplicity Test

When theology sounds complicated, simplicity provides a test: truth rarely needs layers of explanation.

The Sabbath position: God wrote "the seventh day is the Sabbath" in stone. The seventh day is Saturday. Therefore Saturday is the Sabbath. It requires one step and zero additions.

The Sunday position: God wrote "the seventh day" but meant something else. Perhaps the resurrection changed it (nowhere stated). Perhaps the church has authority to change it (human claim). Perhaps God accepts the substitution (nowhere confirmed). This requires multiple steps and multiple assumptions, each requiring independent proof.

One position accepts what God wrote. The other requires explaining why He wrote the wrong day, or why He changed His mind without telling anyone. A genuine message from God would not need three layers of human explanation to become coherent. Jesus demonstrated this principle by stripping away Pharisaic additions: "The sabbath was made for man, not man for the sabbath" (Mark 2:27). The Sabbath argument follows the same pattern: return to what God wrote.

For a detailed response to the "Seal is Holy Ghost, not Sabbath" objection, see Appendix B, Objection 17.

The Priest Wore It First

In ancient Israel, God established a temple system with priests who served as mediators between God and His people. The high priest, the chief religious leader, wore special garments when entering God’s presence. One item in particular foreshadowed the sealing of God’s people in Revelation.

Before John saw angels sealing foreheads in Revelation, before Ezekiel described the man with the inkhorn marking God’s faithful, the pattern already existed in the sanctuary. The high priest entered God’s presence wearing a gold plate on his forehead. Engraved like a king’s signet ring, its message read: "Holiness to the Lord."

"And thou shalt make a plate of pure gold, and grave upon it, like the engravings of a signet, HOLINESS TO THE LORD. And thou shalt put it on a blue lace, that it may be upon the mitre
 And it shall be upon Aaron’s forehead, that Aaron may bear the iniquity of the holy things, which the children of Israel shall hallow in all their holy gifts; and it shall be always upon his forehead, that they may be accepted before the LORD."

Exodus 28:36–38

The word translated "signet" is the Hebrew chotam: a seal ring used to authenticate royal decrees. The high priest’s forehead plate was not decorative jewelry. It was God’s seal of approval on his ministry, the visible stamp of divine authority.

The inscription matters. "Holiness to the Lord" uses the Hebrew word qodesh (holiness, set apart). The Sabbath commandment uses the same root: qadash (to sanctify, to make holy). When God commands "Remember the sabbath day, to keep it holy" (Exodus 20:8), He uses language directly connected to the priest’s forehead inscription.

The Sabbath sanctifies. The plate declared sanctification. The same Hebrew root binds them.

The connection between the Sabbath and God’s seal was first articulated in Adventist theology by Joseph Bates (1792–1872), a retired sea captain who became one of the three founders of the Seventh-day Adventist Church. In 1849, he traced the "sign" language of Exodus 31:13 to the sealing of Revelation 7:3.2 Joseph Bates, A Seal of the Living God: A Hundred Forty-Four Thousand of the Servants of God Being Sealed, in 1849 (New Bedford, MA: Benjamin Lindsey, 1849). Available at Internet Archive: https://archive.org/details/BatesJ.ASealOfTheLivingGod.AHundredForty-fourThousandOfThe. For a comprehensive academic treatment, see P. Gerard Damsteegt, "The Seal of God," Andrews University, https://www.andrews.edu/~damsteeg/seal.html. The high priest bore holiness on his forehead. God declared the Sabbath His sign of sanctification. John saw God’s servants sealed in their foreheads. The pattern was continuous: sanctuary typology foreshadowed end-time reality.

Aaron wore holiness before God’s presence. The Sabbath marks God’s people as holy. The 144,000 stand before the Lamb with the Father’s name in their foreheads (Revelation 14:1). The thread connects Eden to Sinai to Revelation: one continuous testimony written in gold, bound in blue, sealed on foreheads.

The Counter-Mark: The Roman Catholic Church’s Admitted Counterfeit

Daniel’s prophecy, given approximately six hundred years before Christ, identified a power that would attempt to change God’s seal:

"And he shall speak great words against the most High, and shall wear out the saints of the most High, and think to change times and laws: and they shall be given into his hand until a time and times and the dividing of time."

Daniel 7:25

Four key elements in this prophecy:

  1. "Speak great words against the most High": Blasphemous claims to divine authority
  2. "Wear out the saints": Systematic persecution of God’s people
  3. "Think to change times and laws": Attempt to alter God’s law, specifically the time-based commandment
  4. "Given into his hand" for 1,260 years: Temporary but extended period of dominance

"Think to change times and laws." Only one commandment involves time: the Fourth. It is also the only commandment that begins with "Remember," as if God knew this would be the one His people would be pressured to abandon.

Its permanence is confirmed by the fact that all ten commandments (including the fourth) were placed inside the Ark of the Covenant, in the Holy of Holies (Exodus 40:20, 1 Kings 8:9). The ceremonial laws (feast days, sacrifices, circumcision) were written in a book and placed beside the Ark, outside (Deuteronomy 31:26). If the Sabbath were merely ceremonial, God would have positioned it with the temporary ordinances. Instead, He placed it with "Thou shalt not murder," inside His presence, permanent, moral.

Daniel prophesied that the antichrist’s power would attempt to change the one commandment containing God’s seal, the one commandment specifying time, the one commandment identifying the Creator.

History records that the Catholic Church fulfilled this prophecy precisely.

The Roman Catholic Church’s Own Testimony

The Catholic Church doesn’t deny changing the Sabbath. They openly admit it and claim this change as their mark of authority (see chapter 3 for full Catholic source documentation).

They call it their mark: the identifying sign proving ecclesiastical (church institutional) power supersedes biblical command.

The Catholic Church substituted their authority for God’s authority. The seventh-day Sabbath, which God called His seal, they replaced with Sunday worship. They openly call this substitution their mark: the visible proof that their ecclesiastical authority supersedes Scripture.

The change began with Constantine’s first civil Sunday law in AD 321, was enforced by the Council of Laodicea in AD 364 (which pronounced anathema on Sabbath-keepers), and continued through 1,260 years of papal enforcement (see chapter 3 and chapter 8 for full documentation).

When They Tell You Their Mark, Believe Them

When the Catholic Church says "Sunday is our mark of authority," and Revelation warns you not to receive the beast’s mark, the identification isn’t hidden.

They’re telling you what their mark is.

They admit the change. They admit that the Bible doesn’t authorize it. They claim that this proves their authority supersedes Scripture. They call it their identifying mark.

And Revelation 13 warns humanity not to receive the beast’s mark.

The connection speaks for itself.

Why the Day Reveals the Authority Question

The Catholic Church changed more than the Sabbath. Through the same councils and the same Greek philosophy, the Church changed three biblical truths. Understanding all three explains why only one becomes the mark:

What Was Changed Biblical Truth Roman Replacement When
The Day Seventh-day Sabbath (Exodus 20:8–11) Sunday worship Constantine 321 AD, Laodicea 364 AD
The God The Father alone is God (John 17:3) Co-equal Trinity doctrine Nicaea 325 AD
The Nature of Man Soul is mortal (Ezekiel 18:20, Ecclesiastes 9:5) Soul is immortal Adopted from Plato through Augustine

A note on the table above: This book’s core argument concerns the Sabbath. Readers who hold different views on the co-equal Trinity or state of the dead can still engage with the Sabbath evidence. These secondary questions don’t determine whether someone receives the mark. The mark is about allegiance: accepting human authority over God’s command regarding His day of worship. These other changes are included to show a pattern of the Catholic Church substituting tradition for Scripture, not to create additional tests of fellowship.

All three changes came from the same source: Greek philosophy filtered through the Catholic Church’s councils.

All three changes followed the same pattern: Greek philosophy filtered through the Catholic Church’s councils, replacing Scripture. The phrase "immortal soul" appears zero times in the Bible, yet the Catechism of the Catholic Church §366 teaches it as dogma. The co-equal Trinity formulation came from the same councils that banned the Sabbath. (For detailed documentation, see Appendix F on the state of the dead and Appendix G on the Trinity question.)

The mark is specifically about the day, not the co-equal Trinity or immortal soul, because of what Revelation explicitly emphasizes.

Revelation connects the mark to worship.

"If any man worship the beast and his image, and receive his mark in his forehead, or in his hand, The same shall drink of the wine of the wrath of God
"

Revelation 14:9–10

The mark involves public worship allegiance: something visible, enforceable, and economic. Believing in the co-equal Trinity doesn’t show up in your public behavior; you can believe it silently. Believing in an immortal soul doesn’t affect buying and selling; it’s an internal doctrine. But whose authority you accept regarding worship becomes visible through which day you observe. You either keep the day God commanded or the day the Catholic Church substituted.

Only this authority question can be economically enforced. Only this question creates the binary division Revelation describes: those who acknowledge God’s authority to establish His holy day versus those who accept human authority to change it.

That’s why the Catholic Church calls Sunday its "mark of authority." The mark isn’t about a calendar square. It’s about whose authority you recognize when forced to choose. The day makes the authority question visible and enforceable. (For those interested in exploring the Trinity question itself, see Appendix G.)

A deeper question remains. The reason God invested holiness in a unit of time lies in Genesis itself. The first thing Scripture calls holy is not a place, not an object, not a person. It is a day (Genesis 2:3). Every culture builds temples in space. God built His first temple in time, accessible to every human being regardless of geography, wealth, or social standing. The Sabbath is not a relic of ancient religion. It is the original sacred architecture, and the authority question over who may redesign it is precisely the question Revelation places at the center of the final conflict.

The seal of God is the Sabbath (Ezekiel 20:12), the sign of His creative authority. The mark of the beast is its counterfeit, accepting human authority to change what God commanded.

The Enforcement Test: Five Criteria

Not all religious issues qualify as the mark of the beast. Revelation 13 describes specific characteristics that identify what the mark is and how it functions:

"And he causeth all, both small and great, rich and poor, free and bond, to receive a mark in their right hand, or in their foreheads: And that no man might buy or sell, save he that had the mark, or the name of the beast, or the number of his name."

Revelation 13:16–17

For something to be the mark3 The Greek word translated "mark" is charagma (Ï‡ÎŹÏÎ±ÎłÎŒÎ±). Adolf Deissmann documented in Light from the Ancient East (1910) that this term was "an imperial seal of the Roman Empire used on official documents during the 1st and 2nd centuries." The charagma bore the emperor’s name and date, attached to commercial documents as proof of imperial authority. John chose a word his readers would associate with Rome. See Adolf Deissmann, Light from the Ancient East, trans. Lionel R. M. Strachan (London: Hodder & Stoughton, 1910), 341. of the beast, it must meet all of the following biblical criteria:

Criterion 1: A Religious Issue Concerning Worship

The mark isn’t political preference or economic policy. Revelation 14:9–11 directly connects receiving the mark with worship: "If any man worship the beast and his image, and receive his mark
"

The issue is who you worship and when you worship. The Sabbath commandment addresses both: it identifies the Creator and specifies the day set apart to worship Him.

Criterion 2: Involves a Change to God’s Law

Daniel 7:25 prophesied that the antichrist’s power would "think to change times and laws," specifically God’s law. The mark must represent a commandment this power changed.

Nine commandments remain universally acknowledged (even if not universally kept). One commandment, the Fourth, was openly changed by ecclesiastical decree. Only the Sabbath/Sunday question involves an admitted alteration of divine law.

The remaining three criteria confirm the identification. Criterion 3: Sunday laws already exist across Europe and historically across the United States, and ecumenical movements increasingly promote a universal "day of rest" for environmental and social reasons; the infrastructure for global enforcement is present, not theoretical. Criterion 4: "No man might buy or sell, save he that had the mark" (Revelation 13:17). In Eritrea, Sabbath-keeping Christians face imprisonment. In Russia, the 2016 Yarovaya Law restricts Adventist activities. Modern digital payment systems and central bank digital currencies make economic exclusion technically feasible in ways previous generations could not have imagined. Criterion 5: The Sabbath/Sunday question creates the binary division Revelation describes: those who acknowledge God’s authority versus those who accept human authority to change it. No third category exists in the prophetic framework.

No other proposed identification meets all five criteria. Microchips, vaccines, social credit systems, and carbon credits all fail the worship and commandment-change requirements. Only Sunday worship, the Catholic Church’s admitted change to God’s Sabbath commandment, fits every biblical criterion. The clarification at this chapter’s opening bears repeating: the test comes with enforcement, not before. Until then, sincere believers in every tradition worship as they understand.

If Sunday observance is not the mark, two facts have no explanation: the Catholic Church uses the word mark to describe Sunday as proof of their authority over Scripture, and Revelation 12:17 identifies the remnant as those who "keep the commandments of God" in a book warning against the beast’s mark. The seal is the true commandment. The mark is its counterfeit.

Forehead and Hand: The Two Ways to Receive the Mark

Revelation 13:16 describes the mark’s placement: "in their right hand, or in their foreheads." In biblical symbolism, the forehead represents the mind: thoughts, convictions, and beliefs. The hand represents action: what you do regardless of what you believe.4 The interpretation of "forehead" representing intellectual acceptance and "hand" representing practical compliance is a theological interpretation based on biblical symbolism patterns where the forehead consistently represents thoughts/beliefs (Deuteronomy 6:8, Ezekiel 9:4, Revelation 7:3, 14:1) and the hand represents works/actions (Ecclesiastes 9:10, Isaiah 1:15). While this symbolic framework is widely accepted in prophetic interpretation, Scripture does not explicitly define "forehead" and "hand" as belief versus compliance in this specific passage. The core biblical fact remains: Revelation warns against receiving the mark regardless of motivation. [Theological interpretation based on biblical symbolism patterns] Receiving the mark in the forehead means genuinely accepting Sunday worship as God’s will, convinced by tradition and church authority. Receiving it in the hand means complying with Sunday enforcement for economic survival while knowing Saturday is the true Sabbath. Both constitute receiving the mark. Revelation gives no exception for economic hardship: anyone who receives the mark, whether by conviction or compliance, faces the same consequence (Revelation 14:9–11).

This is not about current Sunday-keepers who have not yet understood the Sabbath truth. Sincere Christians following the light they have are not receiving the mark. The mark becomes relevant when Sunday observance is enforced by law and the choice is made with full understanding of what is at stake. Jesus stated plainly: "What shall it profit a man, if he shall gain the whole world, and lose his own soul?" (Mark 8:36).

Current Sunday Law Developments

The infrastructure for Sunday enforcement is not theoretical. Multiple European nations maintain Sunday trading restrictions; Poland strengthened its Sunday trading ban in 2018 after Catholic Church lobbying. Pope Francis’s 2015 encyclical Laudato Si’ connects Sunday rest with environmental healing, framing the day as creation care rather than religious observance. The Sunday law will not come as "worship the beast or die." It will come as compassionate legislation solving multiple crises simultaneously: economic relief, environmental healing, moral restoration, and social unity. Those who refuse will be branded as selfish, unpatriotic, and anti-environment extremists (see chapter 10 for detailed worldwide Sunday law documentation).

God gave the seal. Prophecy predicted the change. The Catholic Church admits the change. Revelation warns of the enforcement. History, Scripture, and current events converge on the same identification.

The mark of the beast is Sunday worship enforced by law.

The seal of God is the seventh-day Sabbath kept by faith.

The question of who enforces the mark, and through what mechanism, is examined in chapter 11.

As stated at this chapter’s opening, God does not condemn people for light they haven’t received:

"And the times of this ignorance God winked at; but now commandeth all men every where to repent."

Acts 17:30

He holds accountable those who’ve seen the truth and rejected it. When the issue becomes clear, when Sunday law forces a choice between God’s command and man’s decree, then ignorance ends and accountability begins.

"Therefore to him that knoweth to do good, and doeth it not, to him it is sin."

James 4:17

This book isn’t condemning your devout grandmother who kept Sunday her whole life without understanding. Her prayers for the family were genuine. Her faith carried her through suffering. Her charity to neighbors reflected the character of Christ. God sees the heart. But this book is warning you, now that you’ve seen the evidence, about a test that’s forming. While Sunday observance remains voluntary, you have time to study, pray, and decide. When enforcement comes, the decision will cost everything.

A further distinction: imperfect practice is not the same as willful rejection. Someone who knows the Sabbath is Saturday, who keeps it faithfully, yet occasionally stumbles (an errand forgotten, a purchase made, or an imperfect rest) has not received the mark. The mark seals those who choose human authority over God’s commandment. Sanctification is a process. The question is direction, not perfection. "If we walk in the light, as he is in the light, we have fellowship one with another, and the blood of Jesus Christ his Son cleanseth us from all sin" (1 John 1:7). Walking in the light doesn’t mean never stumbling. It means knowing where the light is and moving toward it. Your stumbling is not your sealing. Your direction is.

The Seal Is Not a Badge

God’s identifying mark is not a badge you pin on, a box you check, or a performance score to optimize. It is what overflows when a heart aligns with its Creator. The Fourth Commandment says "Remember the sabbath day, to keep it holy," but the holiness lives in the relationship the day was designed to nurture, not in the calendar square itself.

Some who read this have walked into Sabbath-keeping communities and walked out wounded. They encountered scorekeepers who tallied attendance and graded Sabbath activities against invisible rubrics. Those communities had adopted the Pharisees’ inversion: external measurement divorced from internal reality. Jesus warned against exactly this error: "Ye pay tithe of mint and anise and cummin, and have omitted the weightier matters of the law, judgment, mercy, and faith" (Matthew 23:23). But the Pharisees’ failure does not invalidate the commandments they claimed to keep. Their failure was how they kept them. The seal remains true even when its claimants distort it.

God is not tallying your Sabbath attendance. He is inviting you into rest. The seal is not a badge you wear. It is a relationship you enter.

For quick-reference summaries that support this chapter, review Appendix A and Appendix E.