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Over on Politics.Meta.SE a comment by user Guest271314 asserts a repugnant etymology:

...You cannot expect readers to parse when you are engaging in direct communication or "colloquially" speaking. For example, "cheaper by the dozen" is a "colloquial" English term that actually refers to plantation owners forcing prisoners of war to impregnate their mother, resulting in a "dozen" "cheap" children with severe birth defects.

– guest271314 Jan 17 at 18:59

On the other hand, Google Ngram has nothing from the 19th century that remotely suggests such an origin. Its earliest Ngram usage is from The Freemasons' Monthly Magazine, March 1844, and refers to copies of a Masonic handbook for sale.

What is the origin of the phrase "cheaper by the dozen"?

Using Eighteenth Century Collections Online, I found this note at the end of an anti-Catholic pamphlet titled A Protestant's Revolution (Dublin, 1734), where other pamphlets by the same publisher ("S. Hyde, Widow in Dame-street") are advertised for publication. At the end of the list appears a nota bene:

N. B. The above Books are sold cheaper by the Dozen or the Hundred.

The statement appears to appeal to cost in bulk.

It's hard to know when the phrasing became idiom, but this example occurring so early and in the context of selling books suggests that its origins were likely in something as prosaic as what the words literally mean together in a marketing context and not a peculiar plantation breeding program that should be well-documented but yields nothing in the resources I've used so far.

The earliest variant of the phrase I could verify in print was 'cheaper in the dozen', from an article in the Hartford Courant (Hartford, Connecticut) of 24 May 1790 (paywalled, emphasis mine):

In New-York the price [of Webster's Spelling-books] has commonly been thirteen shillings New-York currency a dozen, which is three-pence lawful money cheaper in the dozen....

The exact phrase 'cheaper by the dozen' turns up a dozen years later, in the Hartford Courant (Hartford, Connecticut) of 12 Jul 1802 (paywalled, bold emphasis mine):

N. B. Said Chadwick will sell Morocco [leather shoes] cheaper by the dozen than can be bought at any store in this state.

Considering the context of the early uses, and the semantics of the phrase itself, the origin of the phrase is likely to have been marketing jargon.

With due respect to the fanciful folk etymology you encountered, which in fact deserves only scorn and disrespect, I observe that the phrase was more recently popularized by the 1948 book Cheaper by the Dozen (Frank B Gilbreth and Ernestine Gilbreth Carey), a central theme of which was the putative efficiencies of having a dozen children. Thus, it might be proposed that the title of the book derived from the obviously fictitious practice involving prisoners-of-war.

However, even supposing the authors of the book confessed that the title was sponsored by a scurrilous story, that story is not evidence supporting an origin for the phrase.

I cannot give you an early attestation for the exact phrase "cheaper by the dozen" but the idea is common:

The proper worsted for knitting the stockings is of four-threads, at two-pence an ounce ; but if bought by the dozen pounds, is sold at twenty- four shillings the dozen, which is only three halfpence an ounce ...

from Instructions for Cutting Out Apparel for the Poor etc etc (London, 1789).

In addition to the excellent answers that it talks about bulk savings, the reason why dozen specifically is due to the historical practice of bulk units being twelve. This can be seen in eggs to this day, and also in bakers dozen where the 'base' bulk unit of a dozen is rounded up to thirteen, and 'dozens and dozens' to mean a lot.

So this is a simple substitution of 'dozen' for 'quantity' or 'bulk' to say cheaper in bulk.

This is just a placeholder answer while look for the complete connection to the phrase "cheaper by the dozen".

We know that englishmen in the colonies and what would become the "United States" forced prisoners of war (what englishmen refer to as "slaves") to have sexual intercourse with their mothers, in an effort to produce more prisoners of war, for free labor, fun, glory, material gain, and experimentation, see Medical Apartheid
The Dark History of Medical Experimentation on Black Americans from Colonial Times to the Present by Harriet A. Washington. Forcing young prisoners of war to impregnate their mothers is but one of the horrendous practices of englishmen in the western hemisphere. Without such experiments what is referred to "modern medicine" would not exist. If you find the time, browse the historical medical records of the University of Chicago.

That is the origin of the term motherfucker from which the cheaper by the dozen phrase was used to refer to the offspring of such forced impregnation.

The late Dick Gregory revealed many of these historical facts before their passing. It has been some time since last reviewed the actual phrases usage. Am not entirely certain if Dick Gregory is the one and only source of the meaning of that phrase, within the context of the offspring of male prisoners of war in the western hemisphere being forced to impregnate their mothers to produce more prisoners of war.

It is not surprising that anglophiles deny, or refute such origins of terms in the english language.

Though for an individual who is not an englophile, it is common knowledge that english is first and foremost an equivocal language, intended to be used for deception, capable of having more than one meaning for each word or phrase. There is no such thing as "good" or "proper" english. The language is a result of conquest by the Normans, Romans, whores, convicts, pimps, murderers, imperialism and several hundred years of institutional white supremacy.

In any event, this Dick Gregory - On Slavery, Reparations and Hurricanes

... we can't have no more. How we gonna do this?... So he made me, 14
years old, put a sock over my head, and have sex, fuck my mama...

...cause she didn't want him to get made and start killing the
slaves...

... and so, I had to screw my mama man...

... That's where the word motherfucker comes from and then white
boys shit and they hated the word until they got around black folks...

... and now you go to a white movie and they say mothafucka before
they finish running the credits from the movie...

is a brief introduction to the emergence of the phrase "cheaper by the dozen" being used to refer to the offspring of so-called "black" or african women in the western hemisphere being force to be impregnated by their sons, the result of which was often birth defects, which were still valuable "property" for those englishmen who bought and sold human beings. The interview provides accounts for the origin of the terms honky and motherfucker. Will edit the answer to address specifically how the motherfucker history resulted in cheaper by the dozen.

Note, that same interview describes the origin of the term honky, which is derived from english, or so-called "white" men literally honking the horn of their vehicle outside of the whorehouse they were visiting to buy sex from so-called "black" women.

See also The History Of Male Slave Rape (Buck Breaking) And The Word MotherFucker.

The History Of Male Slave Rape (Buck Breaking) And The Word MotherFucker

...especially in Jamaica, they were doing a process called "buck
breaking" where the white supremacist slave owner or the white
supremacist overseer would literally rape black men in fron tof the
black population in order to break his spirit, in order to break him
down as a man, in order to show dominance against him, in order to
show the rest of the black population that this is not your leader I
just made your leader submit to me sexually.

It was a matter of power and control and domination, and they would
call it "buck busting" but it was really "butt busting".

Slavery continued until 1865... for about two generations, right, so
how do you reconcile that?

You can no longer bring in Africans but you still have the business of
slavery, so they still needed Africans to do the work down south,
especially after the cotton gin was invented

(which Dick Gregory also discloses was based on the inventions of one or more African prisoners of war; whom that Whitney witnessed at or got wind of at plantations. The invention is
attribution solely to Whitney per U.S. Patent, though is still disputed by some today; some credit the invention to a European woman)

So you had the creation of breeding farms. Two of the largest slave
breeding farms in the United States were on the eastern shore of
Maryland, and right outside or Richmond, Virginia. The literally bred
black people like cattle.

They would have a strong black man have sex with a healthy black
woman.That woman could be his mother, his sister, his aunt, his
cousin. It didn't matter. Because the end product was product a child
that I can sell, bodies that I could sell and ship down to
Mississippi, Alabama, Georgia, South Carolina, North Carolina. It was
a business.

We had breeding farms in the United States... Even if you were mother
and son, they would mate you... And so I tell a lot of my patients
and a lot of my friends, you gotta stop using the "MF" word because
that was a description of what actually happened when they would put
the paper bag would go over the head to put one breeder against
another when they were related.

The Origins of The Dirty Dozens

The term the dozens is believed to refer to the devaluing on the
auction block of slaves who were past their prime, who were deformed,
aged or who, after years of back-breaking toil, no longer were capable
of hard labor. These enslaved human beings often were sold by the
dozen. In African American Oral Traditions in Louisiana, African
American author and professor Mona Lisa Saloy writes: “The dozens has
its origins in the slave trade of New Orleans where deformed
slaves—generally slaves punished with dismemberment for
disobedience—were grouped in lots of a 'cheap dozen' for sale to slave
owners. For a Black to be sold as part of the 'dozens' was the lowest
blow possible.”

African American Oral Traditions in Louisiana By Mona Lisa Saloy

"The Dozens" are an elaborate insult contest. Rather than insulting an
opponent directly, a contestant derides members of the opponent's
family, usually his mother. The dozens has its origins in the slave
trade of New Orleans where deformed slaves—generally slaves punished
with dismemberment for disobedience—were grouped in lots of a "cheap
dozen" for sale to slave owners. For a Black to be sold as part of the
"dozens" was the lowest blow possible.