death this old man walked." (Ch. 51)

Ahab seeks the white whale with a monomaniacal desire for revenge. When the first mate, Starbuck, repudiates the idea of chasing Moby Dick, exclaiming,

"Vengeance on a dumb brute...that simply smote thee from blindest instinct! Madness! To be enraged with a dumb thing, Captain Ahab, seems blasphemous." (Ch. 36)

Ahab answers Starbuck's charges, saying that everything that is visible is but "as pasteboard masks." It is behind these masks that lies the unknown "but still reasoning thing[s]." Therefore, man must strike through these masks if he is to know what lies behind them.

"How can the prisoner reach outside except by thrusting through the wall? To me, the White Whale is that wall, shoved near to me. Sometimes I think there's naught beyond. But 'tis enough. He tasks me; he heaps me; I see in him outrageous strength, with an inscrutable malice sinewing it. That inscrutable thing is chiefly what I hate, and be the White Whale agent, or be the White Whale principal, I will wreak that hate upon him." (Ch.36)

Ahab perceives the white whale as a mysterious force of evil that he desires to conquer. He wants to "break through" to the supernatural force that lies behind the physical being of the whale and destroy this if he can.

Ahab seeks revenge for two reasons.

One he has been hunting Moby Dick for years and has not been able to kill him. He is frustrated by his inability to catch and kill  Moby Dick.  He personifies the whale into his mortal enemy.

And two, he lost his leg to Moby Dick so he is even more determined to kill him. 

As mentioned in the previously posted quotation, Ahab seeks "that intangible malignity" that he believes is embodied in the White Whale.  In a separate chapter (42), in fact, Melville considers the whiteness of the whale as symbolic of evil.  Even Ismael, the narrator, finds

It was the whiteness of the whale that above all things appalled me.....As for the white shark, the white gliding ghostliness of respose in that creature....This elusive quality....Now, in allusion to the white, silent stillness of deth in the shark, and the mild deadliness of his habits....Therefore, in his other moods, symbolise whatever grand or gracious thing he wll by whiteness, no man can deny that in its profundest idealised significance, it calls up a peculiar apparition to the soul.

It is this apparition of evil and mystery that Captain Ahab would not only avenge himself, but understand. Why is it white, Ishmael wonders, and why does it

appeal with such power to the soul, and more strange and far more portentous--why, as we have seen, it is at once the most meaning symbol of spiritual things, nay, the very veil of the Christians' Deity; and yet should be as it is, the intensify agent in things the most appalling to mankind. 

It is the "invisible spheres" that Ahab seeks to comprehend in his revenge against the whale.  When Starbuck tell him, "To be enraged with a dumb thing, Captain Ahab, seems blasphemous," Ahab replies,

Hark ye yet again--the little lower layer.  All visible objects, man, are but as pasteboard masks.  But in each event--in the living act, the doubted deed--there,, some unknown but still reasoning thing puts forth the moldings of its features from behind the unreasoning mask.  If man will strike, strike through the mask!  How can the prisoner reach outside except by thrusting through the wall?  To me, the White Whale is that wall, shoved near to me....He tasks me; he heaps me; I see in him outrageous strength, with an inscrutable malice sinewing it.  That inscrutable thing is chiefly what I hate; and be the White Whale agent...I will wreak that hate upon him....Truth hath no confines.

The preternatural White Whale wears the "pasteboard mask" that Captain Ahab is obsessed with striking through.  He would know what metaphysical meaning lies behind this creatures eye that cannot see before him, but only sideways from his head.  Revenge against him for his lost leg is a small part of what Ahab seeks; the Pequod's voyage is a metaphor for life and Ahab is man searching for meaning.

Ahab seeks revenge for having lost his leg; he also has a long scar running down the length of his face. He offers a sixteen-dollar gold piece (quite a lot of money at the time) to the first man aboard who spots the whale.

Ahab makes the mistake of interpreting his injuries in a whaling accident as a personal conflict between the whale and himself.  Not only does he give the whale a name, but he attributes it an anthropomorphic dimension. Moby Dick becomes his personal enemy against "whom" he develops a obsessional grudge.  Paying back Moby Dick by finishing "him" off in an ultimate confrontation becomes Ahab's unique goal in life, as his whole identity and reason for being revolves around this "quest."

Herman Melville, a whaler himself, was a simple man but he had certain insight into human psychology. Through the raging and ranting of his protagonist Ahab, Melville demonstrates that life is what you make of it and that often "problems" are not really circumstantial but rather fixations within the mind. Today Ahab would be a prime candidate for treatment for  obsessive compulsive disorder (OCD syndrome). See the following reference for further information concerning this very particular personality disorder.

Critic Karen Tanguma calls both Ishmael and Ahab "Old World Adams": solitary men who seek the truth on the spiritual depth of the sea, the world and teacher to Herman Melville, who has spent most of his life upon it. As such, Tanguma writes,

Ishmael suffered into knowledge and spiritual rebirth and returned to the human race. Melville enhanced Ahab's unfortunate fall, through the novel's dark elements of evil, fear, and dark history....

In this vast allegorical novel, while Ishmael emerges and grows in wisdom and spiritual knowledge as the sole survivor of the Pequod, he is "escaped alone to tell thee" [Job 1:16] his tale because Ahab ("tragic Adam") becomes entrapped in his dark rage--"oh, lonely death on lonely life"--and he denies God as he seeks to kill the white symbol of spiritual power,

"Towards thee I roll, thou all-destroying but unconquering whale; to the last I grapple with thee; form hell's heart I stab at thee; for hate's sake i spit my last breath at thee."

Ahab is wrapped in the shroud of the sea, having blasphemed Nature and God. Ahab's quest has been one to find meaning, but he has found none.

http://voices.yahoo.com/the-adamic-myth-19th-century-american-literature-1847090.html?cat=38

http://www.enotes.com/topics/moby-dick/in-depth

Ahab's quest is significant for a variety of reasons.  Ahab's quest shows the consuming nature of revenge. Ahab is singular in his focus. He is almost epic in how he sees his purpose as revenge against that which had done him wrong.  When Ahab speaks, this significance is amplified:

Aye, aye! and I’ll chase him round Good Hope, and round the Horn, and round the Norway Maelstrom, and round perdition’s flames before I give him up. And this is what ye have shipped for, men! to chase that white whale on both sides of land, and over all sides of earth, till he spouts black blood and rolls fin out.

There is a perceived absolutism to Ahab's quest.  It defines his very being.  It shows the lengths to which vengeance can overtake the human soul.  This certainty is his focus, and it is something that makes both his quest and his characterization important:  "There was an infinity of firmest fortitude, a determinate, unsurrenderable wilfulness, in the fixed and fearless, forward dedication of that glance.”  For Melville, the "fortitude" that Ahab displays is what makes him so significant and so meaningful to the narrative.

For Melville, the consuming nature of Ahab's quest is significant.  It precludes any balance or sense of reason within him.  Rather, it is seen as "fixed" and binding.  Melville suggests that vengeance as a means to appropriate reality prevents a full actualization of the human being.  It takes away from such development. Ahab's words become evidence of this:  

To me, the white whale is that wall, shoved near to me. Sometimes I think there’s naught beyond. But ’tis enough. He tasks me; he heaps me; I see in him outrageous strength, with an inscrutable malice sinewing it. That inscrutable thing is chiefly what I hate; and be the white whale agent, or be the white whale principal, I will wreak that hate upon him. Talk not to me of blasphemy, man; I’d strike the sun if it insulted me.

The significance of Ahab's quest lies in what it does to him.  Ahab shows that revenge has made him think that "there's naught beyond."  He has become driven by the need to "strike" that which lies beyond his reach. While the whale might represent “Retribution, swift vengeance, eternal malice were in his whole aspect.…," this is also telling as to how Ahab has become as a result of his quest.  

Ahab has lost any semblance or perspective of understanding.  He has become that which is no longer in control.  Anger in the form of revenge is in control of Ahab.  He is no longer in control of it. To this extent, Melville shows Ahab's quest as significant because it is a reminder of how easily the individual can lose their way when matters of pride, revenge, and anger converge within the human soul.  This significance is illuminated in the madness of Ahab, himself:  "I am madness maddened!”   Revenge has become "a vulture feeds upon that heart forever; that vulture the very creature he creates.”  This is where Ahab's quest is significant and where it is meaningful.  It is a statement of what human beings can be when susceptible to revenge, anger, and unchecked emotions.