These two English words are very similar; keep reading for further information.
 

Farther

Farther means "more far/distant" in physical distance.

Tom ran farther than Bill.

Which is farther, New York or Los Angeles?

Farthest means "the most far/distant."

Pluto is the farthest planet from the sun.

Further

Further means "more far/distant" figuratively or non-physically, and can also mean "more/additional":

Nothing could be further from the truth.

Who is further along in her research?

Do you have any further ideas?

Further is also a verb: "to advance":

This class will further your understanding of ethics.

Furthest means "the most far/distant" figuratively.

That is the furthest thing from my mind.

 
The Bottom Line

Many English speakers use both farther and further to mean "more far" in space, but only further can be used for non-physical distance and to mean "more." If you are talking about actual distance, you can usually use either word without raising any eyebrows, but keep further in mind for all of your English furthering efforts.
 

English Difficulties

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