'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.' (188.8.131.52)
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.
'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.' (184.108.40.206)
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.
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. (220.127.116.11)
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.
Penelope's notion is that I should set down what happened, regularly day by day, beginning with the day when we got the news that Mr Franklin Blake was expected on a visit to the house. (18.104.22.168)
Betteredge's daughter, Penelope, has the idea that the best way to tell the story is chronologically. Being systematic about it is the only way to get through these past events and to come to terms with them.
'The stain is taken off,' she said. 'But the place shows, Mr Betteredge—the place shows!' (22.214.171.124)
According to Rosanna, it's impossible to get over the past or to come to terms with it completely. She uses the stain on Betteredge's shirt as an illustration – the stain itself is removed, but you can still see the spot where it was.
I am to keep strictly within the limits of my own experience, and am not to inform you of what other persons told me—for the very sufficient reason that you are to have the information from those other persons themselves, at first hand. (126.96.36.199)
Betteredge finishes his narrative by saying that he mustn't narrate any events that aren't from his own memory and direct experience. Only personal memory is reliable enough to be included in the series of narratives.
In that happy bygone time, I was taught to keep my hair tidy at all hours of the day and night, and to fold up every article of my clothing carefully, in the same order, on the same chair, in the same place at the foot of the bed, before retiring to rest. An entry of the day's events in my little diary invariably preceded the folding up. The 'Evening Hymn' (repeated in bed) invariably followed the folding up. (188.8.131.52)
Miss Clack opens her narrative by describing her memories of childhood. Perhaps she thinks that the precision and detail of her childhood memories will help to assure the reader that her memory of the more recent events surrounding the Moonstone are just as accurate and detailed.
[…] I have continued to fold my clothes, and to keep my little diary. The former habit links me to my happy childhood—before papa was ruined. (184.108.40.206)
Miss Clack uses her mundane habits of folding her clothes and keeping her diary as a way of feeling connected to her "happy childhood" memories of before her father lost his money.
'But I had no happy time to look back at, no past peace of mind to force itself into contrast with my present anxiety and suspense—and I held firm to my resolution through it all.' (220.127.116.11)
Ezra Jennings suggests that having happy memories makes it more difficult to stick to a difficult resolution during hard times, because those happy memories form such a strong contrast to present unhappiness.