Subject: Re: Help Wanted: Constitutional Grammarian
Reading them together, "offices" of the United States are the positions that the President appoints people to - either directly, or that are appointed to those offices by his appointees.

The Appointments Clause of the Constitution says: [the President] shall nominate] "all other Officers of the United States, whose Appointments are not herein otherwise provided for".

This implies that there are Officers whose appointments are provided for by the Constitution. The President is one of those.