Solution: The first thing to realize is that the words in the three columns can be put together to form 7 crossword-style clues, by taking one word from each column. Each solution has 8 letters. The clues and their solutions are as follows:

3D Revolved Ellipse = Spheroid
African American Revolutionary = Malcolm X
Deep Blue Opponent = Kasparov
English King Ironsides = Edward II
Mediterranean Sea Arm = Adriatic
Tailed Nut Lover = Squirrel
Wavelength Measurement Unit = Angstrom


The seven answers all end with a roman numeral. As clued by the diagram, you can line them up by the last letter (in order of roman numeral value) and then look at the letters in the diagonal:

edwardII
kaspaRoV
malcOlmX
squIrreL
adRiatiC
sPheroiD
AngstroM


Reading the letters up the diagonal gives the answer word, something that "needs no explanation", the latin phrase A PRIORI.