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Question: Did Joseph Smith state that the moon was inhabited, and that its inhabitants were dressed like Quakers?
Question: Did Joseph Smith state that the moon was inhabited, and that it's inhabitants were dressed like Quakers?
This is not a quote from Joseph Smith, but rather a late, third-hand account of something that Joseph is supposed to have said
The source for this claim is not Joseph Smith himself; the first mention comes in 1881 in Oliver B. Huntington's journal, who claimed that he had the information from Philo Dibble. So, we have a late, third-hand account of something Joseph is supposed to have said.  Hyrum Smith  and Brigham Young  both expressed their view that the moon was inhabited.
A patriarchal blessing given to Huntington also indicated that "thou shalt have power with God even to translate thyself to Heaven, & preach to the inhabitants of the moon or planets, if it shall be expedient." 
Huntington later wrote an article about the concept for a Church magazine:
As far back as 1837, I know that he [Joseph Smith] said the moon was inhabited by men and women the same as this earth, and that they lived to a greater age than we do -- that they live generally to near the age of a 1,000 years.
He described the men as averaging nearly six feet in height, and dressing quite uniformly in something near the Quaker style. 
So, it would seem that the idea of an inhabited moon or other celestial body was not foreign to at least some early LDS members. It is not clear whether the idea originated with Joseph Smith.
In the 1800s, the idea that the moon was inhabited was considered scientific fact by many
However, it should be remembered that this concept was considered 'scientific fact' by many at the time. William Herschel, the discoverer of the planet Uranus, died in 1822. Herschel argued "[w]ho can say that it is not extremely probable, nay beyond doubt, that there must be inhabitants on the Moon of some kind or another?" Furthermore, "he thought it possible that there was a region below the Sun's fiery surface where men might live, and he regarded the existence of life on the Moon as 'an absolute certainty.'" 
Other scientists announced that they had discovered "a lunar city with a collection of gigantic ramparts extending 23 miles in either direction." 
The 1835 Great Moon Hoax added to the belief in lunar inhabitants
In addition to these pronouncements from some of the most prominent scientists of the day, a clever hoax in 1835 only added to the belief in lunar inhabitants.
John Herschel, son of the famous William, went to South Africa to study stars visible only in the southern hemisphere. This was the cause of considerable public interest, given Herschel's involvement. (William Herschel was the preeminent astronomer of his generation. He had discovered Uranus, and was also of the view that the moon was inhabited. 
On 23 August 1835, Richard Locke published the first article in the New York Sun of what purported to be reports from Herschel's observations. Over a total of six installments, Locke claimed that Herschel was reporting lunar flowers, forests, bison, goats, unicorns, bipedal tailless beavers who cooked with fire, and (most provocatively) flying men with wings:
They appeared to be constantly engaged in conversing, with much impassioned gesticulation; and hence it was inferred, that they are rational beings. Others, apparently of a higher order, were discovered afterwards. . . . And finally a magnificent temple for the worship of God, of polished sapphire, in a triangle shape, with a roof of gold. 
These reports were widely believed and caused a minor sensation. They were carried in the Painsville Telegraph, adjacent to Mormon Kirtland.  The Sun eventually hinted that the matter was a hoax:
Certain correspondents have been urging us to come out and confess the whole to be a hoax; but this we can by no means do, until we have the testimony of the English or Scotch papers to corroborate such a declaration. 
Popular belief in lunar inhabitants persisted for decades after the hoax
No more than this was forthcoming, and the Painsville Telegraph made no mention of the possibility of a hoax. Popular belief in lunar inhabitants persisted for decades. Herschel initially found the episode amusing, but he eventually grew frustrated with having to continually explain to the public that the whole matter was a hoax, with which he had nothing to do: he would later refer "the whole affair as 'incoherent ravings'". 
In a private letter, Hirschel's wife indicated how skillfully the hoax was carried out:
Margaret Herschel was more amused. She called the story 'a very clever peice of imagination,' and wrote appreciately..."The whole description is so well clenched with minute details of workmanship...that the New Yorkists were not to be blamed for actually believing it as they did...." 
Church publications did not shy from embracing later scientific findings on the matter:
Desert News noted:
Proof that the Moon is not Inhabited.
“Dr. Scoresby, in an account that he has given of some recent observations made with the Earl of Rosse’s telescope, says: ‘With respect to the moon, every object on its surface of 100 feet was distinctly to be seen; and he had no doubt that, under very favorable circumstances, it would be so with objects 60 feet in height…. But no vestiges of architecture remain to show that the moon, is, or ever was, inhabited by a race of mortals similar to ourselves….. There was no water visible….”
- “As there is no air nor water on the moon, but very few changes can take place upon its surface. There can be no vegetation and no animals, and although many astronomers have brought their imaginations to bear upon this subject, and have given us descriptions of the beautiful scenery upon its surface, and have even peopled it with inhabitants, we have every reason to believe that it is as barren and lifeless as an arid rock."
Modern prophets and general authorities will sometimes cite newspaper articles or books to illustrate the points which they wish to make. In doing so, they are not endorsing such articles or books as being prophetically correct in all particulars. Rather, they are using the science and information of their day to enhance their preaching of the gospel.
LDS doctrine was not provincial, since it provided for "worlds without number" (Moses 1:33) created by Christ. These worlds held those who would require the gospel, since by Christ "the inhabitants thereof are begotten sons and daughters unto God." (DC 76:24)
Information given to the 19th century Saints by the authorities of the day were consistent with these doctrines, and so they believed them, and occasionally mentioned them in a religious context. As always, prophets and believers are products of their time. Biblical authors, for example, clearly accepted a geocentric (earth centered) cosmos, with a flat earth and heavens supported by four pillars. Like the authors of the Bible, modern prophets are generally beholden to their era's scientific concepts, except where corrections in those concepts are needed to permit the gospel to be understood and applied. This does not mean, however, that prophets of any era do not receive revelation about matters of eternal significance.
- Van Hale, "Mormons And Moonmen," Sunstone 7 no. (Issue #5) (September/October 1982), 13–14. off-site
- Hyrum Smith, "Concerning the plurality of gods & worlds," 27 April 1843; cited in Eugene England (editor), "George Laub's Nauvoo Journal," Brigham Young University Studies 18 no. 2 (Winter 1978), 177. off-site
- Brigham Young, "The Gospel—The One-Man Power," (24 July 1870) Journal of Discourses 13:271-271.
- Patriarchal Blessings Books 9:294–295.
- Young Woman's Journal (1892) 3: 263.
- Patrick Moore, New Guide to the Moon (W.W. Norton & Company, New York: 1976), cited by Van Hale, "Mormons And Moonmen," Sunstone 7 no. (Issue #5) (September/October 1982), 15. off-site
- Van Hale, "Mormons And Moonmen," 15.
- Holmes, 464.
- Moore, New Guide to the Moon 130–131; cited by Van Hale, "Mormons And Moonmen," 16.
- Painesville Telegraph (11 September 1835).
- New York Sun 16 September 1835; cited by Alex Boese, "The Great Moon Hoax," museumofhoaxes.com off-site
- Richard Holmes, The Age of Wonder (London: Harper Press, 2008), 199.
- Holmes, 465, (italics in original).
- Deseret News 6 (1856): 134d.
- ‘Quebec,’ “The Moon”, Contributor 1/9 (June 1880): 193-5, from page 195