How we cite our quotes: Collins doesn't use traditional chapters in The Moonstone, so the citations are a little trickier than in other Victorian novels. Citations follow this format: (Period.Narrative.Chapter.Paragraph).
Quote #1
'In this matter of the Diamond,' he said, 'the characters of innocent people have suffered under suspicion already—as you know. The memories of innocent people may suffer, hereafter, for want of a record of the facts to which those who come after us can appeal. There can be no doubt that this strange family history of ours ought to be told.' (1.1.1.6)
This passage appears on the first page of the novel, and it explains why the story is being told at all. Franklin Blake says that the "memories of innocent people may suffer" if they don't. Franklin is talking about the way these "innocent people" are remembered by others. The best way to make sure that they're remembered the right way is to appeal to the memories of living people.
Quote #2
'We have certain events to relate,' Mr Franklin proceeded; 'and we have certain persons concerned in those events who are capable of relating them. Starting from these plain facts, the idea is that we should all write the story of the Moonstone in turn—as far as our own personal experience extends, and no farther.' (1.1.1.8)
The series of narratives that make up the novel are told from the points of view of people who saw the events as they happened. Only direct experience qualifies a person to tell the story. The narrators are limited to their own memories, only.
Quote #3
When you come to fix your memory with a date in this way, it is wonderful what your memory will pick up for you upon that compulsion. (1.1.3.2)
Betteredge claims that he's able to remember in great detail what happened on a given day over a year before – but only once he comes up with the date. We're not sure about you, but this seems fishy. Let's try it. Can you remember exactly where you were and what you were doing on, say, November 15th of last year? No, we can't either. But hey, maybe Betteredge has an incredibly good memory. For the purposes of the story, of course, we have to assume that he does.