He is remembered in United States history for a clever but harsh comment that he made concerning the dead body of an American officer.

[6] During the American Revolutionary War he was captain of the grenadier company of the Guards Brigade with the rank of lieutenant colonel in the army.

Unlike the other regiments of foot, the light and grenadier companies of the Guards fought with their parent unit rather than being converged into elite battalions.

Colonel Charles O'Hara, who later held important commands in the British army, belonged to the Guards Brigade.

[12] Near Raritan Landing, New Jersey on 31 May 1777, Lieutenant William Martin of Spencer's Additional Continental Regiment and a 20-man patrol were ambushed by 60 Hessian mounted and foot Jägers under Captains Richard Lorey and Carl Wreden.

When the wagon bearing the corpse arrived at Osborn's picket post, he accepted the letter but refused the dead body.

[13] His witty but heartless reply that he, "was no coroner", was received among British officers with great amusement and even made the rounds in London.

[14] At the Battle of Brandywine on 11 September 1777, Mathew's brigade was part of Lieutenant General Lord Charles Cornwallis' division which had reached a position on the American right rear before being detected.

Since Sullivan was responsible for supervising the American right wing, he left the division under the leadership of Brigadier General Philippe Hubert Preudhomme de Borre.

[18] The night before the Battle of Germantown on 4 October 1777, Howe warned Osborn to expect trouble and the detached Guard grenadier and light companies took post on the far right flank next to the Queen's Rangers, a loyalist American unit.

After driving back the American amateurs, the British swung to their left to flank Brigadier General Alexander McDougall's Connecticut Brigade.

Accordingly, he assembled his grenadier company plus an additional four officers, eight NCOs, two drummers, and 150 enlisted men from the Guards Brigade and marched to Province Island on the Delaware River.

In the event, the American garrison evacuated the fort on the night of 15 November and Osborn's detachment occupied the place without opposition.

On 1 September 1795, Prince Frederick, Duke of York and Albany wrote a report to King George III after inspecting military units in the Northern District.