Well, from a slightly different perspective and looking at the fossil record and evolutionary history, this question is quite interesting. Maybe the question should be ‘Why do coleoids have ten arms’, rather than ‘Why does the octopus have eight arms’? Let me explain…..
The basic stock from which all the coleoids are believed to have originated are believed to have ten arms, all of equal length. This is demonstrated in very well preserved belemnites that have ten arms of equal length which was probably also the case with their own ancestors. Those said ancestors quickly split into two other distinct lineages, the Decembranchiata which includes the squid,
Spirula, and cuttlefish. This group slowly modified arm pair IV into seizing tentacles. The other group, the Vampyromorpha, evolved in a different direction, also retaining the ten arms but modifying a different pair, pair II, into filaments which can be seen in the modern
Vampyroteuthis, the only known modern survivor of the group. The Octopus evolved from the Vampyromorphs and lost the second arm pair entirely. The earliest known Octopus from the late Jurassic,
Proteroctopus shows these eight short arms. I attach a diagram, adapted from the family tree to hopefully demonstrate what I am driving at.
To me the real question is how did this ten-armed plan from which all modern coleoids have arisen evolve from the nautilus? The modern nautilus is so different from the other extant cephalopods it is very hard to picture them evolving from that lineage. After all, the nautilus has 90-100 arms, depending on which source one looks at, and an external shell, obviously. It also has two pairs of gills, (the other coleoids have one pair), much cruder eyes, a brain of a different structure to name but a few. These are all fundamental differences let alone the many behavioural differences. Perhaps we are looking at a scenario that the ancient Orders of nautiloids, of which the modern Order Nautilida is the only survivor, were a disparate group with many varied forms of soft-bodied morphologies. The modern coleoids may have developed from a long extinct branch of nautiloid with more modern and familiar features, possibly varying greatly in form from today’s
Nautilus.
Going back to octopuses though, I think I agree with John about selective pressures favouring fewer limbs and more complex behaviour. Think how crude the hunting strategy is for the archaic
Nautilus with over ten times the number of arms, compared to the highly supple and more complex octopus. You could be right about the circle idea, Neptune, works for me!
(Please remember that I am no biologist, and these are just my opinions….)