In America we claim not to convict the accused unless their guilt can be put beyond a reasonable doubt. I believe that there was a time when we actually did this, but we don't now. In fact, when we consider the practical applications of this standard it becomes evident that no reasonable country could function with it in place.
Much of the confusion surrounding the term can be traced back to the common usage of the word "reasonable". Today this word is usually used to connote a sort of general emotional tone, a certain attitude. Thus when we ask that a Jury acquit unless the guilt of the accused can be proven beyond a reasonable doubt, we are instructing them as to what mood they should cultivate while weighing the evidence.
Similarly, when we ask that they regard the accused as "innocent until proven guilty," we bring to the term "proven" a strikingly different definition than we would utilize generally.
When speaking of the "presumption of innocence," we understand that the "presumption" referred to is a "reasonable" one, that is, one which may be waived when it comes into conflict with a particularly shocking case, or with evidence which renders the defendant's guilt especially likely.
The Founding Fathers were intellectuals, philosophers for whom the concept of "reason" had a particularly keen significance. Thus, when they spoke of "reasonable doubt", they were referring to such doubt as might be achieved through the application of reason. Juries were to presume the innocence of the defendant, and thus, wherever possible, interpret the available facts in accordance with this presumption. Thus the defendant would only be convicted if no one could come up with any other explanation to fit the available facts; if the slimmest possibility existed that someone else committed the crime, the defendant would go free.
Of course, in some of the more philosophical areas of doubt, certain assumptions were universally to be made. The always-applicable epistemic issues were generally to be ignored. The testimony of Law Enforcement was generally to be believed, at least as applied to forensic evidence. And, in considering human behavior, all persons were presumed to have behaved rationally unless evidence could be given that they were prone to insanity or hysteria.
This last, more than anything, stands out today. Today we have the psychiatric disciplines. Today we are acutely aware that most people don't behave in anything like a rational way most of the time.
And, as it isn't feasible to assume that those involved acted rationally, we are instead behooved to assume that they acted normally. In this "free country" there is one universally understood restriction: even if you follow the law, you'd better be prepared to explain whatever you're doing to Law Enforcement. You'd better be ready to submit to an on-site interrogation if what you're doing is at all unusual. And God help you if a crime is committed anywhere in the surrounding area. Thus a perfectly reasonable assumption was modified to provide an excuse for the persecution of all who do not conform.
This is only one of the problems with the Justice System in America, but it's a big one. And we are always inclined to forget that, as Ayn Rand said, the individual is the smallest minority of all.