This word means a toilet, especially an outdoor one. The following is from A Dictionarie of the French and English Tongues (1611), by Randle Cotgrave:

Retraict [modern French retrait]: masculine. An Aiax, Priuie, house of Office [= outdoor toilet].

It is a humorous respelling of a jakes, of same meaning, after Ajax, the name of a hero in Greek mythology. The pun rests on the fact that, in the late 16th and early 17th centuries, the name of the Greek hero would have been homophonous with a jakes when stressed on the second syllable. For example, in the following passage from an epigram by the English playwright and poet Ben Jonson (1572-1637), sakes rhymes with Ajax, used in the sense of a jakes:

In memorie of which most liquid deed,
The citie since hath rais’d a Pyramide.
And I could wish for their eterniz’d sakes,
My Muse had plough’d with his, that sung A-IAX.

The word jakes, which is first recorded in the 1530s, is perhaps from the given name Jacques or Jakes, or the genitive of the pet name Jack which has been used in an infinite number of transferred senses; it has in particular been used in names of devices, for example in bootjack and roasting-jack.

The word Ajax in the sense of a toilet is first attested in the late 1590s. It was particularly used by Sir John Harington (1560-1612), an English courtier, author and translator, who invented the first known valve closet, a precursor of the modern toilet. At the beginning of A new discourse of a stale subiect, called the metamorphosis of Aiax: written by Misacmos, to his friend and cosin Philostilpnos (1596), he wrote:

I feare the homely title prefixed to this Treatise (how warlicke a sound so euer it hath) may breed a worse offence, in some of the finer sort of readers; who may vpon much more iust occasion condemne it, as a noysome and vnsauory discourse: because, without any error of equiuocation, I meane indeede, to write of the same that the word signifies.  But if it might please them a litle better to consider, how the place we treate of (how homely soeuer) is visited by thēselues, once at least in foure and twenty houres, if their digestion be good, and their constitution sound.

And in the prologue to the reader, he gave the following mock etymology of a jackes:

Of late yeares, a French Gentleman son to one Monsieur Gargasier […] built a sumptuous priuy, and in the most conspicuous place thereof, namely iust ouer the doore; he erected a statue of AIAX, with so grim a countenance, that the aspect of it being full of terror, was halfe as good as a suppositor: and further to honour him, he changed the name of the house, & called it after the name of this noble Captaine of the greasie ones (the Grecians I should say) AIAX: though since, by ill pronunciation, and by a figure called Cacophonia, the accent is changed, and it is called a Iakes.

In A pleasant Conceited Comedie called, Loues labors lost, written around 1595, the English playwright and poet William Shakespeare (1564-1616) makes Costard, a “clown” (i.e. a rustic), pun on the meanings of Ajax when addressing Sir Nathaniel:

(Quarto 1, 1598)
O sir, you haue ouerthrowne Alisander the Conque-
rour: you will be scrapt out of the painted cloth for this.
Your Lion that holdes his Polax sitting on a close stoole,
will be geuen to Aiax.

A close-stool is a covered chamber pot enclosed in a wooden stool, and Costard introduces this word for the sake of the joke, the arms assigned to Alexander being a lion in a common chair, holding a poleaxe.

 

DOCUMENTATION

 

John Harington gave practical advice on how to build the “Ajax” in An anatomie of the metamorpho-sed Aiax Wherein by a tripartite method is plainly, openly, and demonstratiuely, declared, explaned, and eliquidated, by pen, plot, & precept, how vnsauerie places may be made sweet, noysome places made wholesome, filthy places made cleanly. Published for the common benefite of builders, house-keepers, and house-owners. By T.C. traueller, aprentice in poetrie, practiser in musicke, professor of painting, the mother, daughter, and handmayd of all Muses artes and sciences (1596):

To M. E. S. Esquier.

Sir, my maister hauing expresly commaunded mee, to finish a straunge discourse that he had written to you, called the Metamorpho-sis of AIAX, by setting certaine pictures thereto; […] to instruct you, & all Gentlemen of worship, how to reforme all vnsauerie places of your houses, whether they be caused by priuies, or sinkes, or such like (for the annoyance comming all of like causes, the remedies neede not be much vnlike,) this you shall do.
In the Priuie that annoyes you, first cause a Cesterne containing a barrell or vpward, to be placed either behind the seat, or in any place either in the roome, or aboue it, from whence the water may by a small pype of leade of an inch be conuayed vnder the seate in the hinder part thereof (but quite out sight) to which pype you must haue a Cocke or a washer to yeeld water with some pretie strength, when you would let it in.
Next make a vessell of an ouall forme, as broad at the bottome as at the top, ij. foote deep, one foote broad, xvi. inches long, place this very close to your seate, like the pot of a close stoole, let the ouall incline to the right hand.
This vessell may be brick, stone, or leade, but whatsoeuer it is, it should haue a Current of 3. inches, to the backe part of it, (where a sluce of brasse must stand) the bottome, and sides all smooth: and drest with pitch, rosin, and waxe, which will keepe it frō taynting with the vrine.
In the lowest part of this vessell; which will be on the right hād, you must fastē the sluce or washer of brasse with soder or Cimēt, the Cōcauitie or hollow thereof, must be ij. inches and ½.
To the washers stopple, must be a stemme of yron as bigge as a curten rod, strong and euen and perpendicular; with a strong skrew at the top of it, to which you must haue a hollow key with a woorme fit to that skrew.
This skrew must, when the sluce is downe, appeare through the planke not aboue a straw-breadth on the right hand, and being duly placed, it will stand three or foure inches wyde of the midst of the backe of your seate.
Item, that children & busie folke disorder it not, or open the sluce, with putting in their hāds, without a key, you should haue a little buttō, or scallop shell, to bind it down with a vice pinne, so as without the key it will not be opened.
These things thus placed: all about your vessell and elsewhere, must be passing close plastered with good lyme and hayre, that no ayre come vp from the vault, but onely at your sluce, which stands close stopt, and euer it must be left, after it is voyded, halfe a foote deepe in cleane water.
If water be plentie, the oftener it is vsed and opened, the sweeter; but if it be scant, once a day is inough, for a neede, though twentie persons should vse it.
If the water will not run to your Cesterne, you may with a force of twentie shillings, and a pype of eighteen pence the yard, force it frō the lowest part of your house to the highest.
But now on the other side behold the Anatomie.

This is Don AIAX-house, of the new fashion, all in sunder, that a workeman may see what he hath to do.

Here are the parts set downe with a rate of the pryses, that a builder may guesse what he hath to pay.

A. the cesterne stone or bricke, prise. 0. 6. 8.
B. D. E. the pype that comes frō the cesterne, with a stopple to the washer. 0. 3. 6.
C. a wast pype 0. 1. 0.
F. G. the stem of the great stopple, with a key to it. 0. 1. 6.
H. the forme of the vpper brim of the vessell or stoole pot.
M. the stoole pot of stone, prise. 0. 8. 0.
N. the great brasse sluce, to which is three inches current, to send it down a gallop into the Iax. 0. 10. 0.
And least you should mislike with this phrase I had it in a verse of a graue author, that was wont to walke vp and downe the Court, with a forest bill, I haue forgot how it begun (like a beast as he was) but it ended in ryme.
          O that I were at Oxenford, to eate some Banberie Cakes.
I. the seate with a peke deuaunt for elbow roome, the whole charge 30. shillings eight pence, yet a mason of my maisters was offred thirtie pounds for the like. Memorandum the scale is about halfe an inche to a foote.

Here is the same all put together, that the workeman may see if it be well.

A. the Cesterne.
B. the litle washer.
C. the wast pipe.
D. the seate boord.
E. the pipe that comes from the Cesterne.
F. the Screw.
G. the Scallop shell to couer it when it is shut downe.
H. the stoole pot.
I. the stopple.
K. the current.
L. the sluce.
M. N. the vault into which it falles: alwayes remember that (   ) at noone and at night, emptie it, and leaue it halfe a foote deepe in fayre water.
And this being well done, and orderly kept, your worst priuie may be as sweet as your best chamber.
But to conclude all this in a few wordes, it is but a standing close stoole easilie emptyed. And by the like reason (other formes and proportions obserued) all other places of your house may be kept sweet.

Your worships to commaund T. C. traueller.