FRIDAY, JAN. 22. The flag on the fort informed us this afternoon of the approach of a ship within the rim of our bay. As she neared, the signals on the Dale told her to be an American man-of-war. We conjectured at once that she must be the Congress; but as she rounded into her berth we could not recognize, in her majestic form, the features of our old friend. She proved to be the Independence, commanded by Capt. Lavelette, and bearing the broad pennant of Com. Shubrick. She sailed from the U. States on the twenty-ninth of August, and arrived at Rio de Janeiro in fifty-three days; remained there ten days; doubled the Cape and reached Valparaiso in thirty-four days; stopped there seven, and reached here in thirty-eight. This is splendid sailing; but the Independence is one of the fastest, as well as one of the most powerful ships in our service. Though razeed of her carronades, all her effective force remains. Her battery is a frowning mass of thunder. Her officers are men of enterprise and professional merit. They have brought a mail, well filled with letters and papers, from the United States. If you would know the value of a single letter, let an ocean roll between you and your home. SATURDAY, JAN. 23. The Independence left the Columbus at Valparaiso, under the broad pennant of Com. Biddle, who has instructions to favor us here with a visit. The Columbus was in want of supplies, and would be detained several days in procuring them. She had better lay in all she will require, for there is nothing here. Unless a transport arrives soon, there will not be salt provisions enough on the coast to enable our squadron to go to sea two weeks. There has not been a transport here for six months; our sailors have been living on fresh meat till they hanker for the salt more than they ever did for the fresh. As for clothing, they can hardly muster a shirt a piece, and one pair of shoes among half a dozen is becoming rather a rare sight. This is a hard case, when our markets at home are glutted with these articles. The sailor is required to be faithful to the government, and the government should be faithful to him. He should not be left here barefooted to patter about like a duck in shallow water. It is well for him that it is a California winter through which he is obliged to pass in his destitution; in the same latitude on the Atlantic he would nearly have perished. SUNDAY, JAN. 24. It is difficult to make the Californians understand why you will not attend to office duties on the Sabbath. The apology that you want it as a day of recreation, would be appreciated; but the plea of its sanctity is with many wholly unintelligible. If you would make a person respect the Sabbath, you must rear him in its sacred observance.