What statement from Snowball to Boxer reveals his ruthlessness in Animal Farm?

In Chapter Five of Animal Farm, Boxer accidentally kills a stable-lad during the Battle of the Cowshed and feels instantly remorseful. In response, Snowball tells him:

"War is war. The only good human being is a dead one."

On the one hand, this is Snowball's way of consoling Boxer: he does not want Boxer to feel remorse over a crime which was necessary to prevent the humans from recapturing the farm. Looking deeper, however, this suggests that Snowball is far more ruthless than the reader had anticipated. It shows, for instance, that he is deeply committed to the principles to Animalism, particularly the idea that humans are the enemy, and that he has no qualms about killing a human, should the need arise.  

It is also worth noting the contrast between Boxer and Snowball in their attitude towards the lad's death. It is symbolic of the wider divide on the farm, between the pigs and the other animals, which increases as the novel progresses.

Why does Napoleon feel threatened by Boxer in Animal Farm?

Boxer is the hard working cart horse who is very adaptable to the new philosophy the Napoleon institutes.  In fact, Boxer comes up with a maxim to accept Napoleon's rule, Napoleon is always right, and I will work harder. Like the sincere members of society who believed in the the ideology of socialism, were probably the most annoying to the establishment.

"In the end, once Boxer's health fails and he is no longer able to work, Napoleon sends him to the horse slaughterer. In Orwell's tale, he represents the common working class who unwittingly accept their base existence, because they believe by hard work they will get ahead and that their leaders will protect them. Boxer's lung trouble seems to refer to Orwell's own bouts with tuberculosis."

Boxer is a true believer, he works so hard that he becomes old before his time, thereby becoming useless to Napoleon. The fact that he must get rid of him poses a serious problem for Napoleon, because he does not like the idea of drawing attention to the fact that he must send him to the slaughterhouse.

This flies in the face of the ideology that Animal Farm was built on that all the animals will enjoy a peaceful, relaxing retirement in the grazing field. Napoleon shows the animals, very early on, that retirement for any animal who outlives his usefulness is not going to happen.

Napoleon likes to keep his real purposes veiled, secret, so getting rid of Boxer poses a problem, which highlights Napoleon's real intentions and this must be covered up immediately by Squealer, the chief propagandist.

George Orwell’s Animal Farm is an allegory for the Russian Revolution. Specifically, it’s a story meant to dramatize, in the form of a fable, how the political struggle that was the Russian Revolution went wrong. Every political struggle is a struggle for influence, because influence is easily translated into power.

In this case, Boxer is an influential character among the animals. He becomes the animals’ symbol for hard work and loyalty. Ironically, as he serves Napoleon even to the last ounce of his strength, he endangers his own life because Napoleon begins to fear his influence among the other animals. He doesn’t realize that Boxer is thoroughly loyal, he can only sense his power. So, he has Boxer taken away to the horse slaughterer. Afterwards, he soon finds himself struggling to explain this action when some of the animals realize what has happened.

I'd be interested in knowing what led you to this question. I can't see any reason why Napoleon would be threatened by someone who never asked a question, who exuded loyalty, who was a great worker ... he never seemed like a threat to me. Of course there is always the possibility that a figure as central to the Farm might have tried to lead a rebellion against Napoleon, but that is very unlikely. As pmiranda suggests, the only problem he ever presented was when it was time to "get rid of" him. But even that wasn't a great problem. With Squealer there to create a reality, that the knacker's wagon wasn't what everyone saw it was but that it was an old wagon that needed to be repainted, and with the animals seeming willingness to believe just about anything, even this wasn't much of a problem.

This concept is at the core of the book. Reality isn't as real as we would like it to be (each of us, for instance, has an opinion about George Bush and Barack Obama, even though we really don't know them and only know what someone, probably with an agenda, has told us), and the price of freedom remains "eternal vigilance."

In Chapter Nine of Animal Farm, Boxer injures himself while collecting stone for the completion of the windmill. When word of this accident reaches Napoleon, he has Squealer, his propagandist, tell the animals that he is arranging for Boxer to be treated by a local vet. But the truth is quite the opposite: Napoleon sells Boxer to a horse slaughterer and pockets the money for himself and the other pigs.

When the van comes to collect Boxer, the other animals gather to say goodbye and quickly realise what Napoleon has done. This is a dangerous moment for Napoleon which threatens his position as leader, because every animal on the farm loves Boxer and they are horrified to learn that he has been sent to "the knackers." Napoleon's reaction demonstrates his sense of fear. Through Squealer, he lies to the animals by saying that the vet has recently bought the horse slaughterer's van and has not had time to paint over its sign. Squealer also claims to have been at Boxer's bedside at the time of his death and that his last words were: "Long live Comrade Napoleon! Napoleon is always right." This sounds like something that Boxer may have said, as one of the most loyal and dedicated animals on the farm, and so the others believe Napoleon and Squealer's lies. 

This acceptance of lies removes the threat to Napoleon's leadership. But when the other animals see the pigs drinking whiskey, rumours begin to circulate about where Napoleon got the money from. This episode, then, marks the beginning of the animals' realisation about Napoleon, specifically his tyrannical nature and his plans for the future of Animal Farm. 

By chapter 7, the pigs are firmly in control, even though they are not the strongest animals.  This is because they use the dogs as enforcers.  Napoleon sees Boxer pinning the dog as a threat to his power.

Napoleon and the pigs use the dogs as their private security force.  Napoleon’s power is cemented from the fact that the dogs intimidate the other animals.  When Boxer pins one of the dogs, this is a thread to Napoleon.  Napoleon does not like threats to his power.

When three of the pigs “flung themselves upon Boxer” (ch 7), he defends himself but then looks to Napoleon to tell him whether or not to kill the dogs.

Boxer looked at Napoleon to know whether he should crush the dog to death or let it go. Napoleon appeared to change countenance, and sharply ordered Boxer to let the dog go,whereat Boxer lifted his hoof, and the dog slunk away, bruised and howling. (ch 7)

Napoleon’s countenance (expression) changes because he sees the other animals respond to Boxer’s strength.  He cannot have Boxer as a threat.  The dogs, not Boxer, must remain the threat.

How does Napoleon demonstrate ruthlessness in Animal Farm?

In Animal Farm, Napoleon shows his ruthlessness in several places, most notably in Boxer's death. 

Before sending Boxer off to be killed and turned into glue, Napoleon remorselessly and ruthlessly captures command of power on the farm by chasing Snowball away. This is the first act of Napoleon's approach to totalitarianism and despotism. 

After Napoleon chases his former friend Snowball off the farm, he has countless animals killed who confess to being Snowball's allies.

Though Boxer is no ally to Snowball and is clearly the most dedicated, hard-working and physically capable animal on the farm throughout the revolution, he is betrayed by Napoleon after losing his strength and ability to work. 

Boxer, who has worked tirelessly for Animal Farm, suddenly takes ill. Napoleon announces that arrangements have been made to treat Boxer in a hospital in town. However, the truck that arrives to take Boxer away belongs to a horse slaughterer, and the animals erupt in a great outcry. 

Forcing Snowball into exile and betraying Boxer to his death are both clear examples of Napoleon's ruthless nature.

In Animal Farm, what statement from Napoleon proves his ruthlessness towards Boxer?

There is another quote which illustrates Napoleon's ruthlessness and it can be found in Chapter Seven during a conversation about Snowball. Through his representative, Squealer, Napoleon tells Boxer that Snowball has always been an enemy of Animal Farm:

"Snowball was Jones’s agent from the very beginning — yes, and from long before the Rebellion was ever thought of."

This demonstrates Napoleon's ruthlessness because it shows that he has no problem in telling lies to the other animals and misleading them. Moreover, his determination to blacken Snowball's reputation knows no limits, even though Snowball has not been seen for months and, in reality, poses no threat to Napoleon's power.

Boxer's response to Napoleon's ruthlessness is characteristic of his blind loyalty towards this regime, more generally. He accepts Napoleon's command without any hesitation:

"If Comrade Napoleon says it, it must be right."

This response is significant because it enables Napoleon's ruthlessness to grow since his commands and opinions are never questioned nor challenged. 

Napoleon tells Boxer that he is sending the hard working horse to a rest home when, in fact, he is has sold Boxer to a glue factory where he will be slaughtered. This is obviously ruthless because Boxer has been such an integral part of Animal Farm and the revolution. Boxer is the hardest worker of the animals, to the point that he works himself to exhaustion. Boxer was excessively loyal to Napoleon and followed him without question. To betray Boxer by sending him to a grim death in a slaughterhouse can only be considered cold-hearted and ruthless. However, this mirrors the way many workers in the Soviet Union were treated during the reign of Josef Stalin and that is the point George Orwell, the author of the book, wants to make.

How does Snowball demonstrate ruthlessness in Animal Farm?

Snowball does not get much of a chance to show ruthlessness. About the worst thing he does is go along with the pigs taking the windfall apples for themselves and showing no regret, as the goodhearted Boxer does, that Boxer killed a ranch hand in the Battle of the Cowshed. These, however, are not ruthless actions.

The truly ruthless character is Napoleon, who is based on Soviet dictator Josef Stalin. Napoleon turns on Snowball and runs him off the farm as a traitor, then steals his idea of building the windmill. Napoleon terrorizes the other animals with his pack of trained guard dogs. When groups of animals protest his policies, such as the hens do when he wants to take all their eggs and sell them, Napoleon ruthlessly starves them out. He stages show trials, in which animals confess to crimes they have not committed, such as conspiring with Snowball, and then are executed. He sells the faithful Boxer to the glue factory rather than giving him his promised retirement, even after Boxer has been his advocate and a tireless worker for years and years. Napoleon cares about nothing but his own power and comfort.

How does Napoleon undermine Snowball in Animal Farm?

After the animals take over the farm, conflicts over leadership and administration quickly develop. Napoleon and Snowball have radically divergent visions of the farm’s current management and future direction. Owing to his desire to increase his own power, Napoleon wants to see rapid changes, because these will bolster the animals’ impression of his skilled leadership. Snowball, who is an idealistic intellectual, prefers to plan carefully and invest in programs that are likely to have a long-term benefit. His willingness to share the administrative tasks is not supported by Napoleon, who is worried that Snowball will be more popular among the masses.

Napoleon’s campaign to undermine Snowball begins by disparaging his plan and sowing fear. Whereas Snowball wants to build a windmill, Napoleon tries to scare the animals into thinking they will starve before it is built. When the animals vote in favor, he sends his vicious dogs on Snowball, chasing him off the farm.

The role of Squealer in disseminating propaganda is another crucial element of Napoleon’s plan. Squealer tells the assembled masses that Napoleon actually developed the windmill idea and fully supports its construction. By taking credit for the other pig’s accomplishments, Napoleon projects the impression that Snowball was an ineffective leader.

When the windmill and other projects hit snags, the blame is placed on Snowball—who allegedly sneaks onto the farm and commits sabotage. From casting aspersions on Snowball’s heroism in the Battle of Cowshed, Squealer’s negative false information grows into accusations of treason and secret conspiracy with Farmer Jones.