All 7 Uses of

approach
in
The Adventures of Tom Sawyer

Some vague figures approached through the gloom, swinging an old-fashioned tin lantern that freckled the ground with innumerable little spangles of light.
Chpt 9 *

approached = got near

As the great day approached, all the tyranny that was in him came to the surface; he seemed to take a vindictive pleasure in punishing the least shortcomings.
Chpt 21

"Yes, and you done more than that," said Injun Joe, approaching the doctor, who was now standing.†
Chpt 9

A little green worm came crawling over a dewy leaf, lifting two-thirds of his body into the air from time to time and "sniffing around," then proceeding again—for he was measuring, Tom said; and when the worm approached him, of its own accord, he sat as still as a stone, with his hopes rising and falling, by turns, as the creature still came toward him or seemed inclined to go elsewhere; and when at last it considered a painful moment with its curved body in the air and then came decisively down upon Tom's leg and began a journey over him, his whole heart was glad—for that meant that he was going to have a new suit of clothes—without the shadow of a doubt a gaudy piratical uniform.†
Chpt 14

He climbed over, approached the "ell," and looked in at the sitting-room window, for a light was burning there.†
Chpt 15

And now and then they stooped in a group and splashed water in each other's faces with their palms, gradually approaching each other, with averted faces to avoid the strangling sprays, and finally gripping and struggling till the best man ducked his neighbor, and then they all went under in a tangle of white legs and arms and came up blowing, sputtering, laughing, and gasping for breath at one and the same time.†
Chpt 16

Definitions:

(1) (approach as in: approached the city)
to get closer to (near in space, time, quantity, or quality)
(2) (approach as in: use the best approach)
a way of doing something; or a route that leads to a particular place
(3) (approach as in: approached her with the proposal)
to begin communication with someone about something -- often a proposal or a delicate topic
(4) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

More rarely (and typically only in classic literature), the phrase nearest approach to as used in "her nearest approach to an apology" or "her nearest approach to a smile" typically means that "something is as close to something else as it ever gets." "As near an approach to" can have a similar meaning.