I. Thou, Thee
I grew up in a religiously conservative environment, of the Protestant sort. As this was some decades ago, the preferred Bible version was, of course, the King James. By reading and studying the Bible, and memorizing parts of it, I acquired an intuitive familiarity with the archaic language in use 500 years ago, before I ever read The Lord of the Rings, or heard of Tolkien.
I say "intuitive", because I absorbed the proper grammatical usage without understanding. As "Thou" and "Thee" were used mostly for God, and often capitalized, to boot, I associated them with formal usage, and "you, you" with ordinary usage.
When I first read LOTR, and dived into the appendices, at about age thirteen, I was quite puzzled by Tolkien's reference to "thou, thee" as familiar as opposed to the formal "you, you". It was only when I took high school German, and later Spanish and French, that I realized "thou, thee" were cognates for "du, dich", "tu, ti", "tu, te", all of which are familiar. "You, you" in English were at one time plural or formal, but have gradually displaced "thou, thee".
Why am I putting these pronouns in quoted pairs? Not only that, I have "you, you" as a pair, when they're the same word, right? Not really.
At one time, English, like many other languages, had a case system for all its nouns—they would change form depending on usage in the sentence, whether nominative (subject) or accusative (object). The pronouns are the only ones left that retain cases—"I, me", "she, her", "they, them". "You" however, is used for both nominative and accusative.
Here is where I believe some writers, in an effort to give a sense of intimate, familiar, or even derogatory speech, have misinterpreted Tolkien's commentary on usage.
In LOTR, Appendix F, II On Translation, Tolkien wrote in a footnote:
In one or two places an attempt has been made to hint at these distinctions by an inconsistent use of thou. Since this pronoun is now unusual and archaic it is employed mainly to represent the use of ceremonious language; but a change from you to thou, thee is sometimes meant to show, there being no other means of doing this, a significant change from the deferential, or between men and women normal, forms to the familiar.
When I first read it, I was already familiar with the correct grammatical usage of "thou, thee", so I read the pertinent sentence thus:
...a change from you to [thou, thee] is sometimes meant to show ... a significant change from the deferential ... forms to the familiar.
It seems some authors read the sentence like this:
...a change from [you to thou], [thee is sometimes meant to show ... a significant change from the deferential ... forms to the familiar].
apparently interpreting this sentence to mean "thou" is deferential or formal, and "thee" is familiar.
From my understanding of archaic English, this is not what Tolkien intended. He surely, as a linguist, and expert on the various stages through which our language passed, was intimately familiar with the accurate, historical usage of pronouns in the English of the times when "thou" and "thee" were in use.
I have pulled out a few quotes from LOTR using "thou" and / or "thee".
TT, Book III - Chapter 5 The White Rider
Lock-bearer, wherever thou goest my thought goes with thee. But have a care to lay thine axe to the right tree!
TT, Book III - Chapter 6 The King of the Golden Hall
The king now rose, and at once Éowyn came forward bearing wine. 'Ferthu Théoden hál!' she said. 'Receive now this cup and drink in happy hour. Health be with thee at thy going and coming!'
ROTK, Book V - Chapter 2 The Passing of the Grey Company
The days are short. If thou art in haste, remember the Paths of the Dead.'
The days now are short. Either our hope cometh, or all hope's end. Therefore I send thee what I have made for thee. Fare well, Elfstone!'
Then Isildur said to their king: "Thou shalt be the last king. And if the West prove mightier than thy Black Master, this curse I lay upon thee and thy folk: to rest never until your oath is fulfilled. For this war will last through years uncounted, and you shall be summoned once again ere the end."
'Nor would I,' he said. 'Therefore I say to you, lady: Stay! For you have no errand to the South.'
'Neither have those others who go with thee. They go only because they would not be parted from thee - because they love thee.' Then she turned and vanished into the night.
But she said: 'Aragorn, wilt thou go?'
'I will,' he said.
'Then wilt thou not let me ride with this company, as I have asked?'
'I will not, lady,' he said. 'For that I could not grant without leave of the king and of your brother; and they will not return until tomorrow. But I count now every hour, indeed every minute. Farewell!'
Then she fell on her knees, saying: 'I beg thee!'
ROTK, Book V - Chapter 4 The Siege of Gondor
'What is this, my lord?' said the wizard. 'The houses of the dead are no places for the living. And why do men fight here in the Hallows when there is war enough before the Gate? Or has our Enemy come even to Rath Dínen?'
'Since when has the Lord of Gondor been answerable to thee?' said Denethor. 'Or may I not command my own servants?'
'You may,' said Gandalf. 'But others may contest your will, when it is turned to madness and evil. Where is your son, Faramir?'
'Pride and despair!' he cried. 'Didst thou think that the eyes of the White Tower were blind? Nay, I have seen more than thou knowest, Grey Fool. For thy hope is but ignorance. Go then and labour in healing! Go forth and fight! Vanity. For a little space you may triumph on the field, for a day. But against the Power that now arises there is no victory. To this City only the first finger of its hand has yet been stretched. All the East is moving. And even now the wind of thy hope cheats thee and wafts up Anduin a fleet with black sails. The West has failed. It is time for all to depart who would not be slaves.'
'Such counsels will make the Enemy's victory certain indeed,' said Gandalf.
ROTK, Book V - Chapter 6 The Battle of the Pelennor Fields
A cold voice answered: 'Come not between the Nazgûl and his prey! Or he will not slay thee in thy turn. He will bear thee away to the houses of lamentation, beyond all darkness, where thy flesh shall be devoured, and thy shrivelled mind be left naked to the Lidless Eye.'
A sword rang as it was drawn. 'Do what you will; but I will hinder it, if I may.'
'Hinder me? Thou fool. No living man may hinder me!'
Then Merry heard of all sounds in that hour the strangest. It seemed that Dernhelm laughed, and the clear voice was like the ring of steel. 'But no living man am I! You look upon a woman. Éowyn I am, Éomund's daughter. You stand between me and my lord and kin. Begone, if you be not deathless! For living or dark undead, I will smite you, if you touch him.'
ROTK, Book V - Chapter 10 The Black Gate Opens
'So!' said the Messenger. 'Then thou art the spokesman, old greybeard? Have we not heard of thee at whiles, and of thy wanderings, ever hatching plots and mischief at a safe distance? But this time thou hast stuck out thy nose too far, Master Gandalf; and thou shalt see what comes to him who sets his foolish webs before the feet of Sauron the Great. I have tokens that I was bidden to show to thee - to thee in especial, if thou shouldst dare to come.'
ROTK, Book VI - Chapter 6 Many Partings
Then Éowyn looked in the eyes of Aragorn, and she said: 'Wish me joy, my liege-lord and healer!'
And he answered: 'I have wished thee joy ever since first I saw thee. It heals my heart to see thee now in bliss.'
In the above quotes, some use only one or the other. The shortest that uses both "thou" and "thee" is Galadriel's message to Gimli— Lock-bearer, wherever thou goest my thought goes with thee. But have a care to lay thine axe to the right tree! Note that "thou" is in the position of subject and "thee" in that of object (object of the preposition, rather than of the verb, but it would be the same). If you read through all the quotes, you'll find that this is absolutely consistent—"thou" is subject and "thee" is object.
If you have difficulties figuring out which is which, recast the sentence using "he, him" or "she, her", which are "subject, object" as well. For example: wherever he goes my thought goes with him.
Note also, in the conversations between Aragorn and Éowyn in The Passing of the Grey Company, he uses "you" or "your" in addressing her, that is, polite usage, while she uses the intimate "thou" and "thee". Aragorn keeps her linguistically at arm's length, and she's trying to get closer to him. Contrast this to Aragorn's usage at Éowyn and Faramir's trothplighting, in Many Partings, where he addresses her familiarly with "thee".
Again, in the Gandalf / Denethor conversations, Gandalf consistently uses the polite, deferential "you" (both subject and object), while Denethor talks down to him, using "thou, thee" in a derogatory fashion, as to an inferior or servant. Ditto with Sauron's Messenger.
Interestingly, in the bit about Isildur: Then Isildur said to their king: "Thou shalt be the last king. And if the West prove mightier than thy Black Master, this curse I lay upon thee and thy folk: to rest never until your oath is fulfilled. For this war will last through years uncounted, and you shall be summoned once again ere the end." Isildur switches from "thee" and "thou" when addressing the king, his vassal and inferior, to "you", not as a polite form, but as a plural form, including the king's folk in the curse.
I also went and dug up some Shakespeare, who switches back and forth between "thou, thee" and "you, you", as well.
From Shakespeare, here's Sonnet 18, which uses "thou, thee":
Shall I compare thee to a summer's day?
Thou art more lovely and more temperate:
Rough winds do shake the darling buds of May,
And summer's lease hath all too short a date:
Sometime too hot the eye of heaven shines,
And often is his gold complexion dimmed,
And every fair from fair sometime declines,
By chance, or nature's changing course untrimmed:
But thy eternal summer shall not fade,
Nor lose possession of that fair thou ow'st,
Nor shall death brag thou wand'rest in his shade,
When in eternal lines to time thou grow'st,
So long as men can breathe or eyes can see,
So long lives this, and this gives life to thee.
In his plays, the different usage for familiar / derogatory and polite / plural is also quite evident.
And, from the King James Version:
16 And Ruth said, Intreat me not to leave thee, or to return from following after thee: for whither thou goest, I will go; and where thou lodgest, I will lodge: thy people shall be my people, and thy God my God: 17 Where thou diest, will I die, and there will I be buried: the LORD do so to me, and more also, if ought but death part thee and me.
Of course, throughout the Bible, you can find many instances of "thou, thee".
A curious phenomenon is the "Quaker thee", in which "thee" is used as subject, as well as object, sometimes even as possessive! This was an artificial construct, promoted by a leader of the Quakers, to preserve "plain speaking", or humility and equality in language. They used third-person verb inflections, however—"Thee is" or "Thee has".
II. Thy, Thine
The possessives "thy" and "thine" are similar, and rather easier, once you have a grip on "thou, thee". They work almost like "my" and "mine".
"Is this thy book?"
"No, it is thine. The other book is mine."
There is an exception, which shows up in the Galadriel quote—Lock-bearer, wherever thou goest my thought goes with thee. But have a care to lay thine axe to the right tree! Notice that instead of "thy axe", she uses "thine axe". This is because "axe" begins with a vowel; that is, it follows the same rule as "a, an"—"a book", "an apple". In archaic English, "my, mine" would have behaved the same way. If you are familiar with The Battle Hymn of the Republic, recall that it starts with "Mine eyes have seen...".
III. Doth, Dost
Here's just a bit on verb inflections, as the web resources are many.
In general, "thou" requires the verb to take the "-st" or "est" ending (with a couple of exceptions), while "she", "he", "it" and all other third person singular nouns might lead to "-th" or "-eth". Notice that Tolkien doesn't typically go that far—he limits his archaicisms to the "thou", etc.: Lock-bearer, wherever thou goest my thought goes with thee. But have a care to lay thine axe to the right tree! Here, he uses the archaic "thou goest" alongside the more modern "my thought goes", instead of the archaic "my thought goeth".
The LOTR quotes above use a number of verbs with "thou", most in the "-st" form:
thou art (this is an exception the the "-st" rule)
Thou shalt (also an exception)
wilt thou go? (ditto)
Didst thou think
Verbs from the Shakespeare sonnet, all but one in the "-st" form:
thou ow'st (owest)
and one in the "-th" form:
summer's lease hath all too short a date
That last would be rendered "thou hast" in second person singular.
Once again, of course, both Shakespeare and the King James Bible make great examples of archaic English in actual usage.
I hope you've found this interesting and useful. If you need more information, here are some links.
"A Note on Shakespeare's Grammar"
Includes chart with some typical verb inflections.
More background, grammar, discussion of Quaker usage:
"Plain Speech FAQ"