Machinery
The astute reader will quick learn that machinery is a big time symbol for Arnold. He rages against it and often. Putting faith in the machinery of society is a sure-fire stepping-stone to anarchy. Considering the text was written at a time when the Industrial Revolution was still just warming up, any assumptions that it is intended as a symbol for something much more than actual machines is a good one. Essentially, machinery in the sense Arnold uses it is propagandized reactionary values and assumptions. A more to the point explanation is insert Puritanism whenever the word machinery appears. Which it does a lot.
Barbarians
Barbarians are the symbolic term that Arnold applies to aristocrats of the day. It is definitely worth nothing that the use of the term as symbolism by Arnold does not carry the fully negative weight which it connotes today. Arnold makes the symbolic connection by reminding readers that the barbarian tribes brought much to Europe which was positive and historically transcendent. But the real Barbarian were revolutionaries who did transform progress. By contrast, the aristocracy want to assert and maintain the privilege afforded them by those who came so long before.
Philistines
The symbolism of the Philistines of Biblical times applied to the middle class of Arnold’s time is actually closer in meaning to what modern readers probably conceive when they think of Barbarians. For the middle class as a Philistine is rather slow-witted like Goliath. They see what is in front of them without creativity or sprit and depend upon the machinery of the propaganda they’ve been spoon-fed to make decision lacking critical engagement.
The Greeks
Referred textually by Arnold as Hellenists, the ancient civilization of the Greeks are the epicenter of culture. Symbolically, they represent the ideal of Arnold’s conviction that vulture is about searching for the truth and when it is found committing to see it as it really exists rather than applying a perceptual falsity upon it. Hellenism is also symbolic of the moderation that allows for civilization and culture to both flourish at once.
Hebraism
The symbolic alternative to Hellenistic civilization is that of the Hebrews, here referred as Hebraics. Philosophically, they stand at odds to the Greeks as a result of being a society structured around codified rules of conduct and ritualistic obedience which lacks the zest of the Greeks. On the other hand, Greek civilization fell, collapsed and came under the domination of others whereas the Hebrews have persevered even in the absence of a homeland and persistent persecution through sheer tenacity and will; characteristics which is admired to the point of transformation into symbol.