J. Wagner sends his best respects to the President with the letter from M. Heise & a translation. He has received a letter from Mr. Madison, informing of his arrival with Mrs. M. at Philada. and that Dr. Physic thinks he can relieve her without any operation more painful than keeping the limb in a fixt posture. Mr. Merry proposes to go for Philada. to morrow. Capt. Murray proposes to sail in the Frigate to morrow. The newspapers mention the capture, by boarding, of one of the gun-boats by 3 Algesiras privateers, but that the was released afterwards. Mr. Simpson remained silent on the 7 June about the reported rupture with Morocco.