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Question: What criticisms are related to Oliver Cowdery's 1834-1835 history of the Church?
Critics of the Church conflate Oliver's first and second installments of his Church history in order to "prove" that Joseph was not aware that a "Supreme being" existed three years after he claimed to have had his first vision
When Oliver Cowdery published his version of the history of the Church in December 1834 and February 1835 he did not include a recital of the First Vision story - thus implying that it was not known among the Saints by that point in time. One critical website makes the following claim:
In the first history of Mormonism from 1835 written under Joseph Smith's direction, it says that the night of September 1823 Joseph Smith began praying in his bed to learn 'the all important information, if a Supreme being did exist, to have an assurance that he was accepted of him.' (LDS periodical Messenger and Advocate, Kirtland, Ohio, Feb. 1835). It makes no sense for him to ask if God existed, if Smith had already seen God face-to-face some three years earlier, and knew he existed.
In Joseph Smith's 1835 published history of the church, he claimed that his first spiritual experience was in 1823 after a religious revival in Palmyra that same year. Smith testified that he prayed while in bed one night, to discover if God existed.
These claims, however, are false. Oliver's February 1835 installment did not describe Joseph's First Vision - it described Moroni's visit. It should also be noted that this was not "Joseph Smith's 1835 published history."
Only two years prior to Oliver's history, Joseph's 1832 account of the First Vision clearly establishes the date of both the first vision, and the vision of Moroni
Oliver Cowdery did, in fact, know about the First Vision when he recorded his version of the history of the Restoration—he had physical possession of the Prophet's 1832 history, which contains an account of the First Vision.
In October 1834 Cowdery announced in his newspaper that Joseph Smith would help with the history project but the Prophet himself noted that "no month ever found [him] more busily engaged than November."  In December 1834 President Smith was busy lecturing at the School of the Elders and acting as a trustee for the Kirtland High School and so during this month he sent Oliver a short letter to be included as part of the project, but also noted within it that he learned of his prominent role in the project, and its imminent appearance in the press, by reading Cowdery's periodical! 
- "The First Vision," mormonthink.com.
- Joseph Smith, History of The Church of Jesus Christ of Latter-day Saints, 7 volumes, edited by Brigham H. Roberts, (Salt Lake City: Deseret Book, 1957), 2:170. Volume 2 link
- J. Christopher Conkling, A Joseph Smith Chronology (Salt Lake City: Deseret Book, 1979), 68–69.