Articles home page

What is "prayer" according to the scriptures?

(Garth D. Wiebe, May 2012, last updated Feb. 2016, formerly titled "Broadening what we call 'prayer'")

The English definition of prayer does not exactly match the word in the Bible from which the English word is translated.

Prayer is most popularly and commonly defined and understood as talking or making a supplication to God (or some other deity). So people generally assume that if you are talking to God (or some other deity) then that is "praying," but otherwise, then that isn't "praying." Consequently, we are often accused of presumption when we "command" or "declare" something, instead of "praying" to God to beseech God to do something. But that is because the popular view of "prayer" itself is too shallow and limited, another sacred cow based on the traditions of men.

The root of the Greek word used for "prayer" is ευχη (Strong's G2171, "euche") which means "a vow" (Acts 18:18, 21:23, James 5:15). The verb form is ευχομαι (Strong's G2172, "euchomai") which likewise means to wish/will/vow, and is sometimes translated "pray."

Adding προσ (Strong's G4314, "pros") to that, which just means "toward" (i.e. indicating direction), you get the word προσευχη (Strong's G4335, "proseuche"), which is always translated "prayer," or the verb form προσευχομαι (Strong's G4336, "proseuchomai"), which is always translated "pray."

So, "pray" or "prayer" in the New Testament, as it is usually used, is intrinsically a "toward-wish" or "toward-vow" or "toward-will."

Note that there are two things that are not intrinsically rooted in the original word. The first is "God" (or some deity) and the second is speech or verbalization (whether vocal or silent verbalization). If that were the case, the word would have to look something like "theo-," or maybe "pros-theo-" combined with "laleo" (talk) or "lego" (say) or "eipe" (tell) or "reo" (declare) or "phemi" (aver), to give a sense of verbalization.

So first, you do not necessarily need to pray to God for it to be a legitimate form of prayer.

Second, it is not about an incantation. Jesus says "your Father knows what you need before you ask him" (Matt 6:8), which should be intuitively obvious, since God knows everything beforehand, including the intention of your heart. The speaking part isn't what makes "prayer" prayer, but is a verbal expression of it.

Although prayer normally expresses itself as speech, you can easily get into religious ritual repetition, "babbling like pagans, for they think that they will be heard because of their many words" (Matt 6:7) and so on, whereas it is the vow/wish/will of the heart that God sees, and God is pleased when you combine that with "faith" (Heb 11:6) to make a "vow/wish/will" of "faith," a "prayer of faith," faith being "sure of what we hope for and certain of what we do not see" (Heb 11:1).

James 5:17-18 nicely demonstrates these points, and destroys some sacred cow doctrines (i.e. popular traditions of men) about prayer:

"Elijah was a man just like us. He prayed earnestly that it would not rain, and it did not rain on the land for three and a half years. Again he prayed, and the heavens gave rain, and the earth produced its crops."
What exactly did Elijah pray, and who did he pray to? The book of James, according to James 1:1, is written to "the twelve tribes scattered among the nations," in other words, the Hebrews. The Hebrews had the Hebrew scriptures. If James was going to refer to someone in the Hebrew scriptures, they would be able to look this up. Like the "noble Berean Jews" of Acts 17:11, they would be responsible to "examine the scriptures" to see if what James was saying was true.

In 1 Kings 17:1 Elijah is introduced:

"Now Elijah the Tishbite, from Tishbe in Gilead, said to Ahab, 'As the LORD, the God of Israel, lives, whom I serve, there will be neither dew nor rain in the next few years except at my word.'"
That's the first time we hear of Elijah, and Elijah's prayer was "As the LORD, the God of Israel, lives, whom I serve, there will be neither dew nor rain in the next few years except at my word."

Elijah prays in the name of the "LORD" ("LORD" = Tetragrammaton = יהוה = "YHWH" = "Yahweh," or "Jehovah," equivalent to our saying "in the name of 'Jesus'" today), but he does not address God. Nevertheless, it is clearly a "toward-vow." He says that there will be no rain except at "my word." And who does he "pray" to? He "prays" to wicked King Ahab.

That's at least three sacred cows destroyed: 1. He prays in the name of God, but does not address God. 2. He says that it will be according to his (Elijah's) word, not "God's word." 3. He prays to Ahab, a man.

The second prayer that James is referring to is in 1 Kings 18:41

"And Elijah said to Ahab, 'Go, eat and drink, for there is the sound of a heavy rain.'"
Again, Elijah makes a toward-vow/wish/will of faith. Again, he does not address God. Again he "prays" to Ahab. It was by faith, and not by sight, because he says "there is the sound of a heavy rain" yet there is not a cloud in the sky, and there hasn't been for three and a half years at this point.

In fact, not only does Elijah pay no attention to the sky, he "bent down to the ground and put his face between his knees," so he is now looking at the ground and cannot see the sky. When his prayer of faith (spoken to Ahab) begins to manifest (his servant reports seeing "a cloud as small as a man's hand"), he then warns Ahab to get moving "before the rain stops you."

So, you see that you probably have to broaden your view of what "prayer" is, compared with what you might have assumed based on the traditions of men.

Now, let's go back to look at Elijah's interaction with God at the start of all this:

"Now Elijah the Tishbite, from Tishbe in Gilead, said to Ahab, 'As the LORD, the God of Israel, lives, whom I serve, there will be neither dew nor rain in the next few years except at my word.' Then the word of the LORD came to Elijah: 'Leave here, turn eastward and hide in the Kerith Ravine, east of the Jordan. You will drink from the brook, and I have ordered the ravens to feed you there.'" (1 Kings 17:1-4)
Notice that the word of the LORD came to Elijah after, not before Elijah's "prayer of faith," and it had nothing to do with his "prayer of faith," or rain, or King Ahab, or anything like that. The "word of the LORD" was that he should go hide in the Kerith Ravine.

A lot of what churchianity asks for in prayer is a form of entreating and repeatedly asking/begging for something over and over again, but not being sure what God will do in response, yet hoping God will do something if they ask enough times, or if they can pool resources to get a lot of people praying to God and perhaps fasting to convince God that they are serious (remember Matt 6:7, or the behavior of the prophets of Baal vs. Elijah on Mt. Carmel). But when some of us come in and just declare how something is going to be, commanding sickness to go, the weather to change, or whatever it may be, they take offense at us, as in, "Who do you think you are?". Perhaps they think that only someone like Elijah or Jesus have the authority to do that. But in the case of Elijah, James says he was "just like us" (James 5:17). It's not that we are "just like Elijah" but that Elijah was "just like us." In the case of Jesus, we are supposed to be like Jesus, and we have authority "in Christ."

As an aside, the "like us" of Elijah is ομοιοπαθης ("[h]omoiopathes") which is ομο ("[h]omo"), and means like or same, combined with παθω ("patho"), which also can take the form πασχω ("pascho"), from which we get "patho-logy" (study of disease, i.e. suffering), "paschal" (as in paschal lamb), "passion," "pathetic," and so on. Elijah was passionate about what was wrong in Israel (Ahab, Baal worship, etc.) and decided to do something about it, just like we should be passionate about sickness and sin (James 5 context) and do something about it. In these cases, we "pray" to do something about it, like Elijah did.

Another word is δεησις (Strong's G1162, "deEsis"), which is a "petition." It's not used very much, but in the scriptures, we most often see it here and there in the phrase "prayers and petitions," showing that they are different root concepts (otherwise, why would two separate words be used?).

In that passage of James, we see ευχη ("euche"), προσευχη ("pros-euche"), δεησις ("deEsis'), and ομοπαθω ("[h]omo-patho"). I'll highlight them below (just the noun form, and without the inflections, for simplicity):

"Is anyone suffering evil among you? Let him pray [pros-euche]. Is anyone cheerful? Let him play music. Is anyone infirm among you? Let him call to him the elders of the ecclesia, and let them pray [pros-euche] over him, rubbing him with olive oil in the name of the Lord. And the vow [euche] of faith will be saving the faltering and the Lord will be rousing him up, and, if he should have done sins, it will be forgiven him. Then confess shortcoming-effects to one another and pray [euche] for one another, so that you may be healed. The operative petition [deEsis] of the just is availing much. Elijah was a man of like emotions [homo-patho] with us, and he prays [pros-euchomai] in prayer [pros-euche] for it not to rain, and it does not rain on the land three years and six months. And again he prays [pros-euchomai], and heaven gives a shower and the earth germinates her fruit." (James 5:13-18, CLV, edited)
Note that in this instance (James 5:15) the "vow" (most translations say "prayer") of faith saves the faltering (i.e. makes the sick person well). "The Lord will raise him up." The "vow" is a declaration by a man, and the power to heal comes from the Lord. In this case the "toward-vow" is the prayer of the elders, praying "over" the sick person, with oil, "in the name of the Lord." Note that in this instance (James 5:16) "petition" doesn't mention God as the object, either. Literally it reads "[much] [is-being-strong] [petition] [of-just-one] [in-acting]" so it's about the effectiveness of the petition itself, on account of the righteous/just(ified) one.

There are some other words of interest, sometimes translated "pray" or "prayer," especially in modern English translations.

ικετηρια (Strong's G2428, "[h]iketeria") means supplication or entreaty. It is used only once, in Hebrews 5:7, where the context is about Jesus being the perfect high priest interceding for us. In Hebrews 5:7 he offered "petitions [deesis] and supplications [hiketeria]," showing that petitions and supplications are also different root concepts.

παρακαλεω (Strong's G3870, "parakaleo") combines παρα (Strong's G3844, "para"), which means "near," and καλεω (Strong's G2564, "kaleo"), which means "call" or "bid," forming a compound used over a hundred times, which means to call near, entreat, implore, or beseech. This word occurs only twice in connection with God. The first is in Matt 26:53 where Jesus tells Peter he could entreat God to send twelve legions of angels. But he doesn't. The second is in 2 Cor 12:8, where the apostle Paul entreated the Lord to take the "thorn in the flesh" (a demon) away.

δεομαι (Strong's G1189, "deomai") is the verb form of the noun δεησις (Strong's G1162, "deEsis"), cited previously, which means to beg or beseech, and sometimes translated "pray." It is used 22 times, of which 6 times it is directed at God. In Matt 9:38 and Luke 10:2 Jesus says "beseech the Lord of the harvest to cast out workers." in Luke 22:31 Jesus "besought" that Peter would not fall away. In Acts 8:22,24 Peter rebuked Simon the sorcerer, telling him to beseech God, then Simon asked Peter to do that for him. In Acts 10:2 Cornelius had been continually beseeching God before Peter was sent to him.

εντυγχανω (Strong's G1793, "entugchano") is a verb that means to plead on behalf of or intercede. It occurs in 6 places (including "huper-entugchano" once as a single word), 5 of those 6 in connection with God, 4 of those 5 in connection with the new covenant, of which 2 places are Jesus interceding for us (Rom 8:34 and Heb 7:25) and 2 places are the Spirit interceding for us (Rom 7:26-27).

εντευξις (Strong's G1783, "enteuxis") is the noun form of the above verb that is something pleaded in 1 Tim 2:1 ("...petitions, prayers, pleadings, thanksgiving...") and 1 Tim 4:5 (things being consecrated by the word of God and pleading). in the first case, both "pro-euche" and "enteuxis" are used, showing that they are different root concepts.

ερωτω (Strong's G2065, "eroto") simply means to ask. However, many translations render it as "pray" if it is directed at God. For example, this is the case in a number of instances for Jesus' discussion and long prayer concerning his disciples through John chapters 14-17.

αιτεω (Strong's G154, "aiteo") simply means to request (with an implicit expectation that the request will be granted), either of men or God. Col 1:6 "we do not cease praying and requesting that you may be filled" uses both "pros-euche" and "aiteo," showing that they are different root concepts. In Mark 11:24, "whatever you are praying [pros-euche] and requesting [aiteo], be believing that you are obtaining, and it shall be to you." In Matt 7:9-11 and Luke 11:11-13, this is the word used for the son "requesting" bread/fish from an earthly father, likened to our Father giving us the Holy Spirit in the Luke account. It's used in the "ask...seek...knock" verses (Matt 7:7-8, Luke 11:9-10), and "ask and you will receive" (John 16:24). In John 16:24, "until now you do not request anything in my name. Request and you shall obtain, that your joy may be full" and other similar statements in John chapters 14-16 and Matt 21:22. Asking God for a drink of living water at the well (John 4:10). In Matt 18:19 it is used in the "where two of you should be agreeing on anything" reference. In Matt 6:8 God is aware of what you need before you "request" it. Requesting without doubting of James 1:5-6. In Eph 3:20, God will do exceedingly more than all we are "requesting." In 1 John 5:14-15, we have boldness and will get what we "requesting" because we know God hears us. In 1 John 3:22, we get what we are requesting, because we keep his precepts (believe, love each other). In James 4:3 "you are requesting and not obtaining, because you are requesting evil-ly." Martha says to Jesus in John 11:22 that whatever he requests, God will be giving to him." 1 John 5:16 has one seeing his brother sinning and requesting that he be given life.

βοαω (Strong's G994, "boao"), probably an onomatopoeia, means to implore or cry out. Its verb and noun forms occur 12 times, twice in connection with God. Jesus cries out on the cross "My God! My God! Why have you forsaken me" in Mark 15:34, and in Jesus' parable of the unjust judge he remarks that God will avenge ("ek-dikeo" = [bring] "out-justice" in behalf of) his chosen ones at the second coming, who "implore" him day and night, in Luke 18:7.

In conclusion, we need to take a broader view of what prayer can be. This in no way discounts prayers "to God," where they are appropriate, but just broadens the definition, in line with what the scriptures attest to.

Here are the places where the Greek words occur:

"euche" ("vow"): Acts 18:18, 21:23, 26:29, Rom 9:3, 2 Cor 13:7, James 5:15, 5:16, 3 John :2

"proseuche" ("toward-vow"): All other instances, too many to list, always translated pray/prayer, 140 times in New Testament.

"deesis" ("petition"): Luke 1:13, 5:33, Acts 1:14, Rom 10:1, 2 Cor 1:11, Eph 6:18 (twice), Phil 1:4 (twice), 1:19, 4:6, 1 Tim 2:1, 5:5, 2 Tim 1:3, Heb 5:7, James 5:16, 1 Pet 3:12

"deomai" ("beseech"): Matt 9:38, Luke 5:12, 8:28, 8:38, 9:38, 9:40, 21:36, 10:2, 22:32, Acts 4:31, 8:22, 8:24, 8:34, 10:2, 21:39, 26:3, Rom 1:10, 2 Cor 5:20, 8:4, 10:2, Gal 4:12, 1 Thess 3:10

"hiketeria" ("supplication"): Only in Heb 5:7

"parakaleo" ("entreat"): All instances are entreating other men (106 times), except in Matt 26:52 and 2 Cor 12:8, which is entreating God.

"erotao" ("ask"): 58 instances, too many to list. All are to men, except John 14:16, 16:23, :26, 17:9, :15, :20, 1 John 5:16 (the second part of the verse)

"aiteo" ("request"): 71 instances, too many to list. Most are to men, but some are to God.

"enteuxis" ("plead"): 1 Tim 2:1, 4:5

"entugchano" ("intercede"): Acts 25:24, Rom 8:26-27, 8:34, 11:2, Heb 7:25

"boao" ("implore"): Matt 3:3, Mark 1:3, 15:34, Luke 3:4, 18:7, :38, John 1:23, Acts 8:7, 21:34, 17:6, Gal 4:27, James 5:4

Additional note added on 2/27/2015, edited 2/21/2016

Also, consider the statistics in the following references. These are from the well-known "Perseus" online classical library of Tufts University:

προσεύχομαι ("pros-euchomai," the verb always translated "pray"): http://www.perseus.tufts.edu/hopper/wordfreq?lookup=proseu/xomai&lang=greek&sort=max.

προσευχή ("pros-euche," the noun always translated "prayer"): http://www.perseus.tufts.edu/hopper/wordfreq?lookup=proseuxh/&lang=greek&sort=max.

You'll notice that the vast majority of the instances of the word are in Christian literature, Biblical or otherwise. That is from a vast library of classical literature.

Then look at a reverse-search on the English translated word "prayer": http://www.perseus.tufts.edu/hopper/definitionlookup?q=prayer&sort=freq&target=greek.

Now you will see that προσευχή, for example, occurs 126 times, compared to, for example, ἀρά, which is used 143,667 times (in other words, over a thousand times more frequently). However, the online search engine seems to be picking up all meanings of the spelling of that word, so it is left incomplete as an exercise to see how many of those instances are being used for the word "prayer." What we are looking for is LSJ entry #14990 specifically.

A longer list of possibilities comes out of a reverse-search on the verb "pray": http://www.perseus.tufts.edu/hopper/definitionlookup?q=pray&sort=freq&target=greek.

The point of this is that the Bible usage is very targeted and specific in meaning. The English word "pray" has a meaning that overlaps that, but it is not a match. A Bible "towards-wish/will/vow" is one thing; an English "prayer" is another.

This resolves the dilemma, for example, that we don't "pray for the sick," as the English word is understood; we "heal the sick." Yet we can with integrity say that ours is a "prayer" (a "towards-wish/will/vow") as the Bible defines it, rather than as the English dictionary defines it.

Additional note added on 2/28/2015

One more small point worth making, although you really have to have a good grip on grammar for this: The "-ομαι" ending of the verb, προσ-ευχ-ομαι (Greek verbs are normally cited in lexicons as present tense, indicative mood, active voice, first person, singular number) is the inflection/conjugation that denotes the middle/passive voice of a verb. All the instances of that verb are either middle, or else middle/passive (for the cases where there is no distinction in spelling of the inflection between middle and passive).

Since the active voice of the verb is not used, or no longer used, it is officially called a "deponent" verb form by most scholars. We are officially told to just ignore the fact that certain verbs are spelled as middle/passive, since they are "deponent," and treat them as though they were in the active voice. However, there has been discussion regarding whether there actually even is such a thing as a "deponent verb" (see http://www.dannyzacharias.net/blog/2014/5/16/your-intro-greek-teacher-was-wrong-deponent-verbs-dont-exist and http://jonathanpennington.com/wp-content/uploads/Pennington_Middle_Voice.pdf). Taking that argument into account, which appeals to simplicity, I would disagree with the status quo explanations which form an argument from silence, a fallacy of logic (i.e. that the absence of instances in the literature of an active voice define and prove a "deponent").

For those who are not familiar with "voice," or only so in English grammar, we don't have a "middle" voice in English; the "active" voice denotes something acting upon, and the "passive" voice denotes something acted upon. That is common to both English and Koine Greek grammar. The Koine Greek "middle" voice denotes something that both acts and is acted upon; put a different way, something that acts in such a way so as to be acted upon as a consequence of acting.

The noun ευχη is a "vow," but in the verb form it makes no sense to "be vowed." You can only "vow/wish/will" something; you cannot be "vowed/wished/willed" (unless you are a Calvinist, I suppose). So you will not find a passive-only form of the verb. But neither will you find an active one. Therein is the interesting thing: The act of "vowing/wishing/willing" something commits yourself and invests yourself into the proposition.

If you can get your mind wrapped around this, you will see that προσευχομαι is not just directed at something, like an English "prayer" or the English act of "praying" (to which the assumed object is God). It is "toward-vowing/wishing/willing" something such that you are "toward-vowed/wished/willed" about doing so.

Therefore, the lack of an active voice of the verb teaches us something about "prayer" and "praying" in the Bible. It is not, and cannot merely be a one-way recitation. It is a commitment that you are committing yourself to.

Additional note added on 3/3/2015

Most people are familiar with 1 Thess 5:17

"Pray without ceasing." (KJV)
Does this make sense? Should we be in "prayer" 24/7/365, as "prayer" is defined in English?

Let's first look at the word for "without ceasing."

αδιαλειπτως (Strong's G89, "adialeiptOs"), is a combination of α (Strong's G1, "a"), which is the negative particle meaning "without," just like the English prefix, δια (Strong's G1223, "dia"), which is the preposition/prefix "through," and λειπω (Strong's G3007, also LSJ #64294, "leipo"), which is a verb that means to quit/fail/lack/leave.

The verb λειπω is in the following places:

Luke 18:22 "...still one thing you are lacking. Sell everything..."
Tit 1:5 "...that you should be amending what is lacking..."
Tit 3:13 "...that nothing may be lacking..."
James 1:4 "that you may be...lacking nothing..."
James 1:5 "...if any of you is lacking wisdom..."
James 2:15 "...if a brother or sister should be...lacking nourishment..."
The prepostion/prefix δια adds a sense of continuance, persistence, thoroughness, or permanence to the "lack." διαλειπω (LSJ #25995, "dialeipo," found in other classical literature) means to leave an interval or gap between.

The particle/prefix α negates it.

The -ως ending makes it an adverb.

So the combination is literally "without-through-quitting-ly" or "un-intermittently."

Besides 1 Thess 5:17, in the New Testament the adverb αδιαλειπτως is also in:

Rom 1:9 "...as/how αδιαλειπτως I am making recollection of you always upon the prayers (προσευχων, "pros-euchon") of me.

1 Thess 1:3 "αδιαλειπτως remembering of-you(plural) the work of-the faith and the toil of-the love and the endurance of the expectation of the Lord of-us..."

1 Thess 2:13 "...and we are thanking to-the God αδιαλειπτως..."

The same thing as an adjective (with the -ος, etc., ending), αδιαλειπτος (Strong's G88, "adialeiptos"), is also in

Rom 9:2 "that sorrow to-me is great and αδιαλειπτος pain to-the heart of me"

2 Tim 1:3 "...as I am having αδιαλειπτον remembrance concerning you..."

So you can see that the sense is something that does not quit/cease, or is without a gap in interval, or is un-intermittent.

Coming back around to "praying," 1 Thess 5:17 says,

προσευχεσθε αδιαλειπτως
"proseuchesthe adialeiptos"
"Toward-wish/will/vow without-ceasing."
Here's the thing: The command, at face value, is "without ceasing," "without a gap," or "un-intermittently." That would mean 24/7/365. If this was English "praying," then that would mean no eating, no sleeping, no talking (except to God), and, well, you see the dilemma.

But if it's a "toward-vow" then there is no problem with this verse. You make the vow toward something and then you don't quit the vow.

You don't quit, you don't pull back, you don't back off, from your declaration or intent made in faith.

Here you see that the actual Koine Greek definition of the word makes sense, whereas the English traditional meaning of "praying" does not.

Additional note added on 3/18/2015

This is an additional note about "prayer" in the case of the famous "Lord's Prayer," as it is called, in Matt 6 and Luke 11.

This applies the original point about what προσ-ευχε ("pros-euche" = "toward-vow," the usual Koine Greek word translated "prayer") really means, which is a declaration of intent towards something.

In Luke 11:1, the disciples observed Jesus praying and asked Jesus to teach them to pray. So the response of Jesus was in response to that.

In Matt 6:8, Jesus teaches them "how" (adverb, ουτως, literally "thusly") to pray (not "what" to pray).

In Matt 6:6, Jesus says that our father will απο-δω__ (Strong's G591, literally "give-off," usually meaning "pay" or "render") σοι (to-you), as a result of you "praying." Sometimes this is translated "will reward you," but that is not quite right. The base of the word is the verb "give."

The text of the example prayer is a series of verbs in the imperative mood. The imperative mood, in both English and Koine Greek, expresses an "imperative," a "mandate," a "directive," a "command." In other words, it is not stating something ("indicative mood"), or suggesting something as a possibility or option ("subjunctive mood," i.e. "may" or "should"). In Koine Greek, the spelling of the verb distinguishes these verb moods categorically and unambiguously from one another.

Additionally, in English, the imperative mood is always only in the second person. In other words, the one-word sentence, "Go." has the implied pronoun "you" before it: "[You] Go." In English, taking that pronoun away changes it from present indicative ("You go.") to present imperative ("Go.") However, in Koine Greek, the imperative mood is designated by an inflection (verb conjugation) that can also be in the third person, which cannot be translated easily into English: "[He/she/it/they] Go." The best we can do is add the phrase "Let it be...," as in, "Let it be gone." But that isn't quite right, either, because "gone" is not present tense, and the verb "let" is added. So the bottom line is that you just have to internalize the Koine Greek sense of it.

See this chart of English vs. Koine Greek verb conjugations for an example illustrating these verb conjugations.

Given that, here are the imperatives, which I'll try to translate the best I can, even using improper English grammar and punctuation:

Luke 11:5-8 immediately follows with the illustration of the man going over to his friend's house, where even the friend would not give him the extra bread on account of being his friend, but on account of his αναιδειαν (un-bashfulness).

Luke 11:9-10 immediately follows that with "Request, and it will be given to you, seek, and you will find, knock, and the door will be opened to you..."

Luke 11:11-13 immediately follows that with the illustration of whether a father would be giving him a stone/serpent/scorpion in place of the bread/fish/egg that a son asks of him.

The bottom line is that in these passages Jesus himself defined προσ-ευχε = "toward-vow" prayer by example, illustration, and explanation.

No copyrightI grant this work to the public domain.