Before us is a fascinating Mishnah - Mishnah 4 in the fourth chapter of Maseches Terumos. Let us recall a principle we have already established: a person may not separate Terumah from his fellow's produce. If Reuven separates Terumah from Shimon's grain, the separation has no validity whatsoever - unless Shimon authorized Reuven to act on his behalf.
The Mishnah takes up an interesting question: what is the halachah when a person appointed an agent to separate Terumah on his behalf, but did not specify how much to separate - one in forty, one in fifty, or one in sixty?
The words of the Mishnah and their explanation:
"Ha'omer lishlucho tzei u'srom - torem k'da'ato shel ba'al habayis" - one who says to his agent, "Go out and separate Terumah," the agent separates according to the intent of the owner. The owner of the grain appoints the agent to separate on his behalf, but does not state an amount. The rule is that the agent must act in accordance with the owner's wishes: if he knows that the owner is accustomed to give one in fifty, or one in sixty, or one in forty - that is the amount he must separate. His authority extends only to the amount the owner is accustomed to give.
"Im eino yode'a da'ato shel ba'al habayis - torem k'beinoni, echad michamishim" - if he does not know the intent of the owner, he separates the average amount, one in fifty. When the agent has no information at all about the owner's practice (for example, this is his first day on the job), he must assume that the owner is like the average person, and separate one in fifty.
"Pichais asarah o hosif asarah - terumaso terumah" - if he decreased by ten or added ten, his Terumah is valid Terumah. If it turns out after the fact that he did not match the owner's intent, since had he been asked the owner would have instructed him to give one in forty or one in sixty - this does not matter. Such a deviation falls within the boundaries that the Sages permitted, and the separation takes effect.
"Im niskavein l'hosif afilu achas" - if he intended to add even one:
Here we are dealing with an entirely different case: the agent knows full well how much he is supposed to separate, and nevertheless deliberately decides to deviate from the amount assigned to him - in either direction. For example:
Out of compassion for the poor: the owner is accustomed to give one in fifty, and the agent wishes to add a bit, thinking that his employer can afford it.
Out of thrift: the agent, who serves as the owner's accountant, holds that one in sixty is enough, and wishes to save him money by giving less.
In both cases, even the tiniest deviation made intentionally removes the agent from the authorization he was given. Since he did not act by force of that authorization, he has no power at all to separate Terumah, and his separation is null and void. He acted on his own initiative and according to his own plans, and not in his role as the owner's agent - and an agent is nothing more than the extended hand of the one who sent him. Since he did not act in that role, he is like someone who is not an agent at all, and someone who is not an agent cannot separate Terumah from another person's grain.
It is worth noting the precise wording of the Mishnah: the agent "intended to add" - to add to the denominator. Instead of one in fifty he gives one in fifty-one, so that he has added to the number and reduced the amount of Terumah. But there is no difference between the directions: whatever such an agent takes does not acquire the halachic status of Terumah, since he did not act by force of the mandate he was given.
And if by some wonder the matter comes to light - for example, the agent admits to the owner of the grain that for some reason he did not follow the usual practice - the entire act is retroactively nullified, and one must go back and start again from the beginning.
In summary: in this Mishnah we learned that an agent appointed to separate Terumah without being told the amount separates according to the owner's intent; and if he does not know the owner's intent, he separates the average amount - one in fifty. An unintentional deviation, in the range of ten in either direction, does not invalidate the Terumah; but a deliberate deviation, even the slightest one, removes the agent from the scope of his authorization, and his separation is entirely void.