What does the phrase By April mean in this sentence?
By April 1857 the army had grown to approximately 700 men.
Does it signify before the end of April or at the beginning of April?
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The author is counting by months. April is not only a period of time, but a counter of time; here you are counting time by months; thus the discrete unit of time is not meant to be broken down. Here April means the month of April in its entirety. It's like counting by 5s: 5, 10, 15, 20, 25. There are no units in between and there is no conception of at the beginning of or at the end of 15 (for example).
I would always take such a statement to mean by sometime during April or before April was out.
Had they meant before 1st April, or by some date in April, they would have likely said something else, such as before April 1857 or by 12th April 1857.
I would take it to mean "Before or at the beginning of April".
Think of it logically: imagine that someone tells you that they want a particular job done "by April". This is a bit vague: you don't perhaps know exactly when in April they will check that the job is done. The only way to make sure that it's done when they check it is to make sure that it's done before the first of April.
It's a similar thing in the example: imagine we were to look in a history book and discover that the army grew massively on the 10th of april. We could then say to the speaker "I checked, and on the 1st of april, the army was still relatively small. So I think your statement isn't correct". They can't really have any reasonable defense in this instance: they have made a vague and misleading statement.
The only way the writer/speaker can avoid this accusation is to make sure that their statement is true for the whole of April, which means that it needs to have happened before the start of April.
"Does it signify before the end of April or at the beginning of April?"
Most probably the writer is being imprecise because s/he probably doesn't know when the 700 figure was attained other than it was before the end of April.
Consider that information on this troop movement was likely gained by reading old letters, extrapolating numbers, trying to reconcile inconsistent reports. Also note that it says "approximately 700 men", no doubt with the awareness that people were coming and going all the time, and no one was keeping track that closely.
With this in mind, simply saying "by April" is probably the best choice, as it gives a time frame but doesn't imply more precision than exists.
By: Prep. If something happens by a particular time, it happens at or before that time. In this case, "By April 1857...",(as the writer didn't specify the exact date, it could be any date within April), the army had grown to approximately 700 men. another example could better explain this, "I will get this work done by today." it means this work would be done by anytime before 12:00pm today. In a similar way, "by April" would mean any possible date within April.