On January 10 the Constitution Center published their 10 fascinating facts presidential inauguration. The Constitution Center is a secular, non-partisan, federal government sponsored, organization that relies on an advisory panel of expert historians and scholars. Therefore we have reason to take them seriously as a reliable source of information about American history. They originally said this:
Author Washington Irving claimed George Washington started the tradition of adding “So help me God” at the oath’s end. There is no direct evidence of that. Others believe Chester Alan Arthur used the words when he took the oath in private after James Garfield died.
President Arthur took his presidential oath of office twice, the first time in his apartment in New York. A short but detailed account of that first, unplanned, oath recitation that names six participants was published the next day in the Omaha Daily Bee. That report quotes the oath and clearly states that there was no SHMG, see http://chroniclingamerica.loc.gov/lccn/sn99021999/1881-09-21/ed-1/seq-3.
After traveling to Washington D.C., Arthur was inaugurated again two days later by Chief Justice of the United States Morrison R. Waite in a larger ceremony held in the Vice President's office with invited guests. It was widely reported in newspapers, starting early the next day, that he added the phrase "so help me god" to his oath of office during that second oath of office recitation. The published description of what happened included the who, when, where, and what details that require an eyewitness to reveal. Participants in the second ceremony as described in the newspaper article, several dozen of whom were identified by name, included two associate justices of the Supreme Court, two former presidents, cabinet members, some Senators, and some Representatives.
I do not know who wrote the article or if subsequent accounts claiming Chester Arthur said SHMG rely on this initial report or independently confirm it. Nevertheless, the publish article is credible enough to be accepted absent any eyewitness subsequently contradicting it. Sometimes the evidence favors a conclusion that so and so said such and such and it is misleading to suggest otherwise, even though technically it is true that we lack certainty.
In contrast, Washington Irving's biography of George Washington fails to qualify as an eyewitness account of the inauguration. Irving did not identify an eyewitness, he himself was too far away in the crowd from the president elect to be the eyewitness, he was six years old at the time and he published his biography over 6 decades after the fact, and his account of the inauguration was copied from an earlier account written by someone else who was an eyewitness without acknowledgement or permission [Memoir of Eliza S. M. Quincy, no SHMG in that eyewitness account]. Also, that SHMG story is counter-evidenced by a contemporaneous eyewitness account of the oath recitation written by someone who stood near the president elect on the balcony [letter of the French consul, Comte de Moustier, April 30, 1789]. If George Washington appended SHMG then not only did no one notice, but for the next ninety two years, starting with George Washington's second inauguration, no one else did that again, even though the other presidents repeated the first inauguration's public ceremony with hand on the bible followed by kissing the bible that was required by New York state law for the first inauguration held in NY.